Assembly Puzzles
               

The Assembly or Put-Together class includes those puzzles which entail the arrangement of pieces to make specific shapes in either two or three dimensions, to mesh in a particular way (without necessarily interlocking) or to fill a container or tray. The pieces are free to be juxtaposed in many different configurations but only one or a few arrangements will be valid solutions. The best such puzzles often permit a seemingly valid construction of all but one of the pieces, where the last piece stubbornly just won't fit! For the most part, the order in which the pieces are put together does not matter - when order does matter, it is a sequential assembly puzzle. If the pieces truly interlock to form a free-standing construction that remains stable in various orientations, the puzzle belongs in the Interlocking class. If the pieces have designs on them and there are rules about how the designs must appear, check the Pattern class or the Jigsaws class.
 

Packing Puzzles

Simply stated, the challenge of a packing puzzle is to fit a given set of pieces into a container. The boundaries are either enforced by walls and a lid, or sometimes just walls, with the "lid" implied by the requirement that no piece extends beyond the level of the walls. The container might also be more of a tray, especially if the pieces don't stack in 3 dimensions.

Now, if you consider this task in the abstract, the entire container could be construed as implied rather than physical, and then many assembly puzzles could be considered to be packing puzzles. For example, the SOMA cube could be re-cast as "fit the pieces into a cubic box." In addition, you can shoehorn dissections in here by thinking of the original form as the "container" - the objective is to re-construct the original form, which is tantamount to fitting the pieces back into this abstract container.

For my purposes here, I will include a puzzle in the "packing" category if there is a physical container, and some pieces to cram into it. In rare instances the container is similar to the pieces themselves. Sometimes the puzzle is presented with a subset of all the pieces except for one of them packed into the container, with seemingly no room for the additional piece, and the objective being to rearrange the pieces to make the last piece fit, too.

Take a look at Erich Friedman's Packing Center.

Bill Cutler has written an interesting essay on box packing puzzles.

In addition to his seminal designs of Interlocking puzzles, Stewart Coffin has designed many great packing puzzles. When Coffin's designs appear in the tables below, I have highlighted them like this.

Single-Layer Packing Puzzles with Identical or Similar Pieces


Hercules - B&P
Designed by Jean Claude Constantin
Nice quality and poses just the right amount of challenge.

Crazy L
A very nice little packing challenge, from the Puzzle and Craft Factory.

Four T's - Binary Arts/Thinkfun

Pack the Tray (8 triangles + 1 rectangle) - Saul Bobroff
I got this prototype from Saul at the 2009 NYPP.

Houses and Factories
Designed by Richard Hess - distributed by B & P

Foxes and Wolves
Designed by Richard Hess. Purchased at IPP 29 in SF.

Lucky 7 - Melissa & Doug

Blockade - B&P
Blockade is like Lucky 7 - both use 3 small and 4 large L shaped pieces, but Blockade also has pins on the board and corresponding holes in the pieces. Lucky 7 is trivial to solve - Blockade adds a little (but not much) challenge.

Butterfly - Nature's Spaces
Fit nine identical penta-hexes into a triangular frame. Only one arrangement will work.

Frog Pond - Nature's Spaces
Fit nine identical tetra-hexes into a triangular frame.

3 Ls
Fit the 3 L-shaped pieces into the tray.

Snake Pool
Eleven cubes are loosely strung along an elastic to form a cube snake. Fit the snake flat in the tray - the "pool." There are at least four different solutions. The cubes are 3/4", the tray opening is 3.25" square.
The snake configuration is: 3+2+2+2+1+1 (where a + denotes a right-angled bend that can swivel).
Erich Friedman shows various square-in-square packings on his Packing Center site, but I don't think the solution shown for 11 squares works with this particular cube snake configuration.

Packing Quarters - B&P

Kinato
Kinato is a very nicely packaged puzzle from Ravensburger. Sixteen triangles are threaded via clever swivel connections. Arrange them into a large triangle with the proper pattern. I found it at jigsawjungle.com.

Ampelmann - Roman Götter and Peter Knoppers
Purchased from Roman at IPP31 in Berlin
A black case with a hidden complex interior and two circular openings.
Three red "Don't Walk" Ampelmann figures, and three green "Walk" figures (one mirror image or the other two), colored on only one side.
Two challenges: 1) place all six figures in one "compartment" with one red Ampelmann in the middle, and 2) place all six figures in the other "compartment" with one green Ampelmann in the middle.
The clear piece is a hint - it shows the shape of the cavity inside the case. Simple, eh?
These figures are the old East Berlin crosswalk signal symbols - one of the few vestiges of Communist rule that Berlin citizens want to keep. Read more about " Speciation and Competition in Berlin's Traffic Lights."

Mimi packing puzzles: A, F, H

Modest Hexominoes by Dr. Richard Hess (IPP17)
Place all 20 pieces so that each hexomino shape contains five identical pieces. Includes a booklet with 100 additional problems to maximally cover polyomino shapes with congruent tiles.

The Massai packing puzzle from Siebenstein Spiele, 2011.
Pack the 5 identical L-shaped tetrominoes in the tray.
My copy might be defective, but I found one solution and my wife and kids found two more.

Quartet in F - Stewart Coffin (#253)

Octet in F, designed and made by Stewart Coffin, exchanged at IPP32 by Rosemary Howbrigg

FN Puzzle - pack the four pieces in the tray in three different ways
designed by Mitsuhiro Odawara
produced by Toshiyuki Kotani
Purchased at IPP32

Retrofit - designed by Goh Pit Khiam,
made by Eric Fuller,
from Rosewood, Ipe, Walnut, Bubinga,
Padauk, Maple, and Acrylic.
The following tray-packing puzzles were all designed by Edi Nagata.
Edi sells versions in 2-sided trays, made from MDF. A couple were offered by Bits and Pieces with wooden 2-sided trays and aluminum pieces, other single-sided versions in CD cases by Embrain via Torito.

Pencil Case

Shirt Case
Purchase the 2-sided MDF version from Edi, or the single-sided CD-case versions "Shikoku" and "Australia" from Torito. Philos offers a version, too.

Arrow Case
aka Packing Arrows - B&P

Cat Case
aka Cats in a Cradle - B&P

Cup Case

Baby Ducks Case

Single-Layer Packing Puzzles using a Set of Related Pieces

This is a special group where the pieces aren't identical, but they are related by some rule or theme, which distinguishes them from those puzzles in the more generic group having an assortment of dissimilar pieces. Some of the puzzles in the latter group may languish there though they belong in this section because I am unaware of the rule relating the pieces...

One event at the International Puzzle Party (IPP) is called the Edward Hordern Puzzle Exchange. Qualifying attendees can sign up to participate - each must submit a new puzzle design, and if approved, bring enough copies of the puzzle to exchange one with each other participant (up to 100). IPP32 in Washington D.C. in 2012 was the first time I participated in the exchange. There were 79 puzzles in the exchange in 2012.

For the exchange, I created a tray-packing puzzle I call Non-Convex Bi-Half-Hexes. Catchy and mellifluous, eh? I chose to use a subset of the hexiamonds as pieces. If one divides a regular hexagon in half along a line connecting opposite vertices, then re-joins the two halves along a side-length, there are only seven resulting shapes that are non-convex. Using the "standard" piece names, this set of seven includes { butterfly, chevron, crook, hook, snake, sphinx, yacht }. The other five hexiamonds are either convex { rhomboid, hexagon }, or not composed of two half-hexes { crown, pistol, lobster }. I used this set of seven (mathematically complete given the defining rule) and designed four different simple symmetric trays into which all seven pieces can be packed flat (allowing gaps), three of which have only one solution apiece. The puzzle was produced by Steve Kelsey at AccurateLaserEngraving.com. The case cover is made from a single piece of wood, with a clever laser-cut flexible "binding." We designed a nice dovetail closure.

From what folks tell me, this is a difficult puzzle. If you are interested in purchasing a copy, please email me at the address on my home page.

 
Non-Convex Bi-Half-Hexes, designed by Robert Stegmann
produced by Steve Kelsey at AccurateLaserEngraving.com.

I have been working with Steve Kelsey to design an expansion set for my IPP32 exchange puzzle Non-Convex Bi-Half-Hexes. I have come up with a dozen new tray shapes into which all seven pieces will fit flat. Difficulty ranges from easy to hard, with several having only 1 or 2 solutions, but a few having 7, 9, and 12 solutions. All of the trays are "hollow" and require no internal islands. Our intent is to make the original puzzle and the expansion set available for purchase via Steve's Accurate Laser Engraving website. Stay tuned!


Nine Squared - Tom Lensch
All nine pieces have identical thickness but each has a different combination of length and width selected from discrete increments within a narrow range. When arranged correctly into the tray they simply drop in and out with no binding. Several incorrect packings seem like they should fit, if only you press down a little... wrong!

Apothecary's Cabinet - Constantin
(purchased at GPP)
Each "drawer" has a combination of side tabs and portions of the row separators, and is equivalent to a rectangle with each side having either a tab or a notch. There are 2^4=16 possible arrangements including rotations and reflections. The knobs on the drawers require the reflections. The fact that the side tabs/notches are off-center requires the rotations. This puzzle is a nice realization of a 4x4 heads/tails edgematching puzzle, but includes a cabinet/tray/frame which constrains the solution, since it has all notches along the left and top, and all tabs along the right and bottom. If you assign a 4-bit binary ID to each drawer using 0 for a notch and 1 for a tab, the low bit for the top and high for the left side, then one solution is:
15 7 5 9
14 4 8 13
10 6 1 12
11 2 3 0

For issues 61 and 62 (Nov 2003) of the CFF newsletter, Dieter Gebhardt wrote articles analyzing this puzzle, and in issue 62 reports results derived by Jacques Haubrich.


Digits - Constantin
Fit the 10 digits into the tray.

Partridge Puzzle by Robert Wainwright
obtained from Robert at the 2007 NYPP
Kadon offers some of Erich Friedman's "Partridge" puzzles.
In an "anti-Partridge" puzzle, there is one largest piece, and the count goes up as the pieces shrink.

Square Dance - designed by Derrick Schneider
Purchased from Pavel Curtis - I've been wanting one for a while and was pleased to find Pavel had resuscitated it!
Square Dance won an Honorable Mention in the 2002 IPP Design Competition.
There is only one way to join two 2x2 squares by a half edge,
and only four ways to join a third 2x2 square by a half edge to the first two.
These are the four pieces of Square Dance, and there is only one way to pack them into an 8x8 tray,
and only one way to pack them into a 7x9 rectangle. The included tray is two-sided.

Lonpos Cosmic Creatures

The much-copied Digigrams, designed by Martin Watson.
Made by Eric Fuller, from Grandillo, Walnut, and laser-cut acrylic.

Pentagon Tiles, designed and exchanged at IPP32 by Rene Dawir, made by Marcel Gillen

13 Triangles, designed and exchanged at IPP32 by Ed Pegg Jr., made by William Waite

Di-Half-Hexes, designed and exchanged at IPP32 by Peter Knoppers, made by Buttonius Puzzles & Plastics
I was really surprised to see this one, since it is so similar to my Non-Convex Bi-Half-Hexes IPP32 exchange puzzle. What are the odds? Peter and I must have been hit by a similar brain wave. Fortunately, his puzzle uses a different set of hexiamonds and different trays.

Triangle Edges - designed by William Waite in 2005
Pack the 12 pieces into the tray.
The puzzle is based on a triangular grid
and each piece is composed of five edges.

Single-Layer Packing Puzzles using an Assortment of Dissimilar Pieces


Karin's Star Cluster
An entry in the IPP24 Design Competition.

Tessellating Galaxies - JVK

Sun Dance - JVK

The City
2001 Binary Arts (Thinkfun)
Pack six heptominoes (3 distinct pieces and their mirror images) in the 6x7 tray. Nice metal pieces with 3D abstract buildings on them which prevent the pieces from being flipped and exclude most of the otherwise possible 80 assemblies.

Geometrex Set - Ormazd, Nabucho, and Quirinus
In each case the pieces can be rearranged within the tray to fit in an extra square.

Fit To A Tee - Thinkfun
A nice metal tray-packing puzzle from Thinkfun. Pack the 9 pieces representing golf holes complete with tees, sand traps, and pins, into the base. The base presents a challenge on each side (the front and back nines), with different arrangements of fixed water hazards to work around. Oh, and just as on a real course, abut each flag with the tee of the next hole!

Fantastic Island

The IQ Link puzzle from Smart Games designed by Raf Peeters

The "845 Combinations" puzzle is almost like pentominos... here is a solution to the 845 puzzle.

On 6/22-25/13, made a trip to Niagara Falls. At Niagara Falls, I stopped in at Turtle Pond Toys. They carry several nice puzzles of various types. I picked up the IQ Puzzle from Toyland Company. It has 9 curved pieces that must be fit into the channels within a 4x4 grid of circles. It is an easier version of the 845 Combinations puzzle, which has 10 pieces to be fit into the same grid.

Adam's Cube

One Way

Boxed In - Milton Bradley

Circle Challenge - Melissa & Doug
A good one for kids - work on it from the inside out. The pictures on the pieces are merely decorative.

Magic Block (MCS promo)

Figa Block

IQ Block

Double Cross - Mag Nif
There are four pink plastic pieces and the tray. The objective is to form a cross (plus sign) in the tray.

Sleazier - Pavel Curtis
based on Stewart Coffin's Four Sleazy Pieces (#169A)
Fit the 4 polyominoes into the tray. IPP25

Stewart Coffin's Sunrise / Sunset (#181)
Fit the 5 polyominoes into each side of the tray, making a symmetric pattern in each case. Gift from Bernhard Schweitzer (thanks!). IPP22

Stewart Coffin's Drop In (#153B) aka The Trap
Fit the four pieces into the box through a small slot. They must be arranged so all fit within the inside perimeter of the box walls. Saul Bobroff IPP23

Stewart Coffin's Few Tile (#133)
Made by John Devost
A beautiful Padauk frame about 5.75" squared, with corner splines, and Birch plywood pieces.
A gift - Thanks, John!

Stewart Coffin's Four Fit (#217)
Made by Tom Lensch. Purchased from Tom at the Dartmouth College Mechanical Puzzle Day in Feb. 2008.

Stewart Coffin's Five Fit
From Dave Janelle at Creative Crafthouse.
Fit the five pentomino pieces into the square tray.
The tray has a handy storage space for one of the pieces
should you be unable to solve it.

A Stewart Coffin Tray Puzzle Set (#181), in Poplar and Lyptus woods, made by Tom Lensch. Purchased at PuzzleParadise.ca. This set includes six of Coffin's tray-packing puzzles - a single-sided rectangular tray (#181, 1 solution), a two-sided pentagonal tray (#181-C, The Housing Project, 1 solution each side), and another two-sided pentagonal tray having a movable wall segment on one side (#181-A, The Castle Puzzle, 3 solutions; #181-B, The Tree Puzzle, 2 solutions, other side #181-B, The Vanishing Trunk Puzzle, 1 solution).

Stewart Coffin House Party (#250)
Fit the four Marblewood pieces into each side of the tray, which is made from Poplar on Baltic Birch.
Made by Tom Lensch

Stewart Coffin's Cruiser (#167)
Made by Walter Hoppe.

Lean 2, designed by Stewart Coffin, made by Tom Lensch, exchanged at IPP32 by Dave Rossetti

Buridan, designed, made, and exchanged at IPP32 by Vladimir Krasnoukhov

Heart and Bud, designed, made, and exchanged at IPP32 by Yoshiyuki Kotani
I received these laser-cut tray packing puzzles from Steve Kelsey. Check out Steve's online shop at Accurate Laser Engraving. Steve offers designs you just cannot easily find anywhere else! Thanks, Steve!

Steve has produced a set of Stewart Coffin tray-packing designs, each cleverly housed in a laser-cut wooden "book" case, with a flexible "binding."

Few Tile (closed)

Few Tile (pieces)

Four Fit (pieces)

Four Sleazy Pieces (pieces)

Engelberg Square (pieces)
Designed by Nick Baxter

Think Square - Pressman
There are 4 small right triangles, 4 large right triangles, 4 stair-case shaped pieces, and 5 small squares. The pieces can be fit snugly into the tray with and without one of the five small squares.

Triadenspass - Logika

Pack It In - Great American Puzzle Factory 1996
Pack a set of 16 items into a suitcase frame. Flat cardboard pieces.

The Trapped Man - Tom Jolly
Laser cut by Walter Hoppe. Five unusually convoluted pieces, including the little "man." The first challenge is to fit them into the tray so that none can slide or rotate. Next, try it with only four of the five pieces, then with only three! Several other puzzle goals accompany the Trapped Man puzzle.

Pac-Man - Milton Bradley
First create 4 Pac-men with open mouths. Then use the same pieces to create 3 Pac-men with closed mouths. There are eye stickers on some pieces, which must be positioned correctly. The pieces can be flipped.

The Jayne Fishing Puzzle - A 1916 advertisement of Jayne's Tonic Vermifuge (yuck!). Discussed in Slocum and Botermans' "The Book of Ingenious and Diabolical Puzzles" on page 15. You were to cut out the fish and the ring and then pack the fish inside the ring. The fish names are (left to right, top down): Codfish, Shad, Red Grouper, Cowtrunk Fish, Flying Fish, Bluefish, Mackerel, Tarpon, Sheepshead, Moonfish, Striped Bass, and Weakfish.
Also see No Fishing by Bepuzzled.

No Fishing - Bepuzzled 1998
Remove the water then fit all twelve fish into the bowl. This is a nice wooden laser-cut, colorful, and faithful copy of the Jayne Fishing Puzzle of 1916.

In the Raging Rapids puzzle from Thinkfun (Binary Arts), you have to fit all the men into the raft, facing the right way. The figures' bases have various patterns of tabs and indents.

In the Mayan Calendar puzzle from William Waite, you have to fit all the glyphs into the tray, facing the right way. The glyphs have various patterns of tabs and indents. (Similar to Raging Rapids.)

Alex Randolph's Moebies - Springbok 1973
There are 8 sockets at various positions in the orange board. Six pieces and six pegs are included - the object is to find a way to peg the six pieces to the board so that all fit within the edges.

Springbok Fitting & Proper
Here is a nice set of small, pocketsized tray packings designed by William Waite, purchased from his PuzzleMist website:

From left to right, they are: Triangle Quorn, Square Quorn, Hex Quorn, Diamond Teaser, and Mix Teaser 2.


The Kitchen Ceiling Puzzle - designed by William Waite in 2006
Pack the 12 pieces into the tray so that the holes make symmetric patterns.

Optimal Tumble - designed by William Waite in 2010
Pack the 12 pieces into the tray so that the holes make symmetric shapes.

Vintage 1969 packing puzzles from Lakeside. So far they include:
  • 16 Trains and Planes
  • 18 Horses and Riders
  • 19 Animals
  • 20 Cars and Trucks
  • 21 Fish and Birds

JVK Tessellating Hexagons

Galaxies & Stars - JVK

"Tripple 7" - 3-piece packing (prototype) - JvK

Easy Eight / Hard Eight - Bob Hearn

Wetten Dass...
Also known as FACT
Purchased in Berlin.
The tray has a moving bar, pivoted at one corner. When the bar is aligned along the top edge, the five pieces are easy to pack into the tray. When the bar is aligned along the side edge, it's more difficult.

Toysmith 11 pc. wood puzzle

Mind the Gap - Chris Morgan
Some tray packing puzzles designed by Naoyuki Iwase (Osho) - Mouse, Tulip, and Seals:

See Osho's website, Puzzle-In.


eLeL4 - designed by Hiroshi Yamamoto
presented at IPP30 by Hiroshi Uchinaka
Fit the four pieces into the 8x8 tray. Each piece is composed of 2 'L' shapes.

Unique U - designed by Hiroshi Yamamoto
Fit the six U-shaped pieces into the 9x9 tray.

The Nifty Fifty from Jean Claude Constantin requires you to pack the four pieces into the tray.

The Quartet Puzzle - the Quartet's tray has a movable end wall, and you must pack specific subsets of pieces into the tray depending on where the wall is positioned.

Four in a Frame - a two-sided four-piece tray packing puzzle based on a triangular grid, designed by Markus Götz

Pack-Man - Chris Enright

Hexagon 10

Game Ball Puzzle

Animals of Australia
Dump out the ten nicely cut
animal pieces and try to fit them back in the tray.
No peeking at the solution!

Forever Wild - Animals of the Adirondacks
Pack the ten nicely laser-cut animals into the tray. The animals all go in with a specific side upwards.
From Creative Crafthouse.

Forever Wild - Animals of the Appalachians - a tray packing puzzle, one of a series - each comprising a high-quality tray and nice laser-cut pieces in various colors.

The Cook's Cupboard Puzzle
Pack the eleven kitchen items into the tray.
From Creative Crafthouse.

Hexus, a packing puzzle from Brainwright. Seven pieces and a movable "challenge block" to be placed on a hexagonal grid according to a series of 44 challenges. Purchased at Necker's.

Quadrillion - designed by Raf Peeters, produced by SmartGames
Thanks, Raf!
Arrange the four base plates per a challenge, then pack the pieces on.

Packing Squares

This section describes several types of puzzle in which assortments of square pieces or tiles must be packed in various ways. Much study and analysis has been done in this area, and there are some great resources on the web. Topics include:

Mrs. Perkins' Quilt

The problem of Mrs. Perkins' Quilt (or Mrs. Perkins's Quilt) appeared as no. 173 in Henry Ernest Dudeney's 1917 book Amusements in Mathematics. You can find the book and the problem online in a few places, including at www.gutenberg.org, and at www.scribd.com.

The problem: given a large square quilt made of 13x13 small squares (169 small squares total), find the smallest possible number of square portions of which the quilt could be composed - i.e. a dissection of the large square into a number of smaller squares that don't all have to be different. However, only prime dissections are allowed - the side lengths of the component squares cannot all have a common factor - they must be relatively prime. There can be no sub-square which is itself divided - such a solution is termed "primitive" - primitive quilts are quilts without sub-quilts.

Martin Gardner devotes chapter 11 in his 1977 book Mathematical Carnival to Mrs Perkins' Quilt and Other Square-Packing Problems. Ed Pegg discussed the problem on his Math Games site. The problem is also discussed at mathworld.wolfram.com.

The solution comprises 11 squares and is shown at gutenberg.org. It contains the following number of squares of given sizes: 1x72, 2x62, 1x42, 2x32, 3x22, and 2x12.

The smallest numbers of squares needed to create relatively prime dissections of an n×n quilt for n=1, 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (Sloane's A005670).

Karl Scherer discusses additional variations at his website. Karl defines a nowhere neat tiling - in which no two tiles have a full side in common, and a no touch tiling - where tiles of same size cannot touch, noting that no-touch are always nowhere-neat.

Squared Rectangles and Squares

The problem of Mrs. Perkins' Quilt leads to other questions. In general, how might it be possible to dissect various rectangles or squares into smaller squares? Such puzzles are known as Squared Rectangles and Squared Squares. If a dissection results in pieces all of different sizes, the dissection is called perfect, otherwise it is imperfect. If the dissection does not contain any smaller square or rectangle that is itself further divided, it is called simple (or primitive), otherwise it is compound.

The order is the number of tiles used.

When describing solutions, it is convenient to use a notation called Bouwkamp code. One lists the side lengths of the tiles as they appear in the solution, in left to right order, top to bottom, bracketing groups with flush tops.

There is a nice article in Martin Gardner's 1962 book More Mathematical Puzzles and Diversions, in chapter 17: Squaring the Square - by William T. Tutte, from Gardner's November 1958 column in Scientific American.

Stuart Anderson of New South Wales has a great website called www.squaring.net where he discusses this topic in depth, and gives lots of historical information. Some of the diagrams below are adapted from Stuart's site.

The topic is also discussed at mathworld.wolfram.com.


In 1925, Zbigniew Moroń (1904-1971), of Wraclow, Poland, published a paper, 'O Rozkladach Prostokatow Na Kwadraty' (On the Dissection of a Rectangle into Squares), in which he showed a simple perfect squared rectangle (SPSR) of order 9. Reichert and Toepkin (1940) proved that a rectangle cannot be dissected into fewer than nine different squares (see Steinhaus 1999, p. 297).

I have the plastic Perfect Squares (Le Carre Parfait) puzzle by Dollarama (China). It's got 9 pieces to be packed into a tray. I measured the tray cavity and the piece dimensions, and allowing for measuring error, manufacturing tolerances, and gaps so the pieces can be easily manipulated, this is an example of the Moroń 1925 SPSR.

 

IdealActual (mm)
tray32x33158x163
11887
21573
31468
41047
5944
6838
7734
8419
915


Simple perfect squared squares (SPSS) begin at order 21. Here is A.J.W. Duijvestijn's 112 from 1978:

In Bouwkamp notation, the Duijvestijn 112 is symbolized as:

[50, 35, 27], [8, 19], [15, 17, 11], [6, 24], [29, 25, 9, 2], [7, 18], [16], [42], [4, 37], [33]

The number of simple perfect squares of order n for n >= 21 are 1, 8, 12, 26, 160, 441, 1152, ... (Sloane's A006983).

For a compound perfect squared square (CPSS), the lowest order is 24. This square was found in 1946 by Theophilus Harding Willcocks. The fact that it is the lowest-order example was proved in 1982 by Duijvestijn, p. J. Federico and P. Leeuw.

The highlighted area is a rectangle that is further sub-divided - its presence makes this a compound solution.

Partridge and Anti-Partridge Puzzles

Robert Wainwright presented the Partridge Puzzle at the second Gathering for Gardner, in 1996.

Partridge puzzles call for the dissection of a large square into a set of smaller squares, without voids, such that one small square tile of size 12 is used, two of size 22 are used, three of size 32 are used, up to n of size n2. Kind of like the "Partridge in a Pear Tree" song, the number of square tiles of each size increases by one at each step.

They're based on the following mathematical equivalence:

1 x 12 + 2 x 22 + 3 x 32 + ... + n x n2 = 13 + 23 + 33 + ... + n3 = (n(n+1)/2)2

Bill Cutler, using a variation of his BOX program, found that the smallest value of n for which a packing exists is 8, that there exist 2332 distinct order-8 solutions, and that there are no order-7 solutions.

Ed Pegg has an interesting article on Partridge puzzles on his Mathpuzzle site. There's also some information at Erich Friedman's site. Kadon sells some of Erich Friedman's Partridge puzzles.

Here is an order 8 puzzle I bought from Robert Wainwright at the 2007 NYPP:


Erich Friedman also discusses Anti-Partridge tilings. In an Anti-Partridge Puzzle, one must dissect a square using n copies of a 1x1 square, (n-1) copies of a 2x2, (n-2) copies of a 3x3, etc., through 1 copy of an nxn. They're based on the mathematical equivalence:

n x 12 + (n-1) x 22 + (n-2) x 32 + ... + 1 x n2 = k2

There exist solutions for (n,k) of: (1,1), (6,14), and (25,195)...

The (6,14) square was found by Colin Singleton in 1996.

Packing a Series of Squares (Gaps Required)

Another type of square-packing problem, discussed by Ed Pegg Jr., is to find the minimal side m of square m2 into which one can pack one of each square of sides 1, 2, 3, ..., n. In this problem, there can be voids. In fact, in this type of problem packing the large square without gaps is not possible. The only series of squares which sum to a square is for squares of sides 1 through 24, which sum to 702 = 4900. (This is also the only number that is both square and pyramidal - i.e. 4900 balls can make a square, and also be stacked in a square-based pyramid with layers of 1,4,9,16, etc. - proved by G. N. Watson in 1918.) A proof that no perfect tiling of the 702 with squares 1-24 exists was done in 1974 using exhaustive computer search by Edward M. Reingold (Gardner 1977).

The Sloane sequence A005842 gives a(n) = minimal integer m such that the m2 square contains all squares of sides 1, ..., n.

This problem has practical applications, such as electronic circuit layout.

Minami Kawasaki gives a catalogue of known solutions.

From Ed Pegg, here is a packing of 1-51 into a 214x214:

The Calibron Twelve Block Puzzle

The Calibron Twelve Block Puzzle was made by Calibron Products of West Orange, N.J. ca. 1932. I don't have one, but the dimensions of the pieces are shown on Iwase's site. I've been intrigued by this puzzle for some time and I thought I'd cover it here.

If you search Google Books for calibron puzzle, you will find links to an ad for the puzzle, selling for $1, in the Jan 1935 issue of Popular Science magazine, an entry for the puzzle in the Catalog of Copyright Entries showing the puzzle was copyrighted on Dec. 22 1932, and an ad in the 1933 New Yorker Vol. 9, claiming that the puzzle has "Baffled over 900 scientists at a recent convention."

About.com says that the company Calibron Products was "established by Theodore Edison (1898-1992) [Wikipedia] [bio at nps.gov] to keep some of his late father's employees and engineers working together on research projects." Theodore's obituary in the New York Times on Nov. 26 1992, says he was the last surviving child of the inventor Thomas Alva Edison.

From the inside of the box: "The problem is to arrange the twelve blocks to form a single large rectangle. Any rectangle will do, provided that all twelve blocks are used... We guarantee that there is a straightforward, accurate solution of this puzzle in a single plane, and without recourse to any kind of trick... However, in spite of the enormous number of possibilities, there appears to be only one basic arrangement which satisfies the above conditions... We once offered $25 for the first solution of this problem and distributed hundreds of puzzles at that time, - but recieved almost no correct arrangements! We should like to hear from you if you succeed in making the rectangle unaided."

Here is a list of the 12 pieces, using halved dimensions:

  • 1) 32x11
  • 2) 32x10
  • 3) 28x14
  • 4) 28x7
  • 5) 28x6
  • 6,7) 21x18
  • 8,9) 21x14
  • 10) 17x14
  • 11) 14x4
  • 12) 10x7

Why not buy or make a set of pieces and try this puzzle yourself, before looking at the solution hidden here?

 

 

(This space intentionally left blank.)
 


Josh Jordan was kind enough to send me a copy of the Calibron 12-Block Puzzle remake produced by Pavel Curtis. Thanks, Josh!
For the remake, Pavel has used my halved dimensions.
Pavel has posted an interesting write-up of the Calibron puzzle.
You can buy a nice wooden version at Creative Crafthouse.

Prime Squares and Cubing the Cube

Carlos Rivera, on his website www.primepuzzles.net, poses an interesting question about "prime squares" - Is there any SPSR or SPSS having only tiles with prime-number side lengths?

The answer is no. Arthur Stone proved that in a perfectly squared rectangle (or square), with at least two square elements, at least two elements have even sides. His proof is on pages 149-150 of "Squared Squares: Who's Who & What's What" by Jasper Dale Skinner, II, published in 1993. ISBN: 0963656902.

Here is another negative result...

While messing about with planar tilings, it's natural to think about extending the problem into 3 dimensions. Can a cube be dissected into a finite set of distinct sub-cubes? The answer is no.

This problem is discussed in Martin Gardner's article, and also online in an article by Ross Honsberger.

Proof: Assume a packing of a cube using a finite set of distinct sub-cubes can be done. The bottom layer will contain a set of cubes, and one of them will be the smallest in that layer. That smallest cube cannot be along an outside edge - i.e. touching a side of the container (other than the bottom) - because if it was, then there would have to be an even smaller cube next to it. Think about it - there are two cases: either it would be in a corner, against an outside wall and with a larger sub-cube next to it, or along an edge with a larger cube on either side of it. In either case, one side of the smallest cube is bordered by walls extending past it. So, any cube that could fit against it must be smaller than it, which violates our premise that it is itself the smallest in that layer.

That means it must be somewhere in the interior, bordered on four sides by a larger sub-cube. That, in turn, means that its upper face must be completely walled in (again, think about it - every bordering cube is larger than it is, but they're all lying on the same plane as it, so the sides of all its neighbors rise above its upper face). That means that its upper face has to be covered by a set of even smaller cubes. Now, if you think about this state of affairs, you'll see we can start all over again with the previous logic - that covering set itself must contain a smallest member which cannot be on an outside edge...

This goes on indefinitely, requiring an ever-smaller set of sub-cubes, and proving that the original assumption is false.


Now, this doesn't mean we can't have fun in 3 dimensions...

Yukiyasu Sekoguchi has designed many puzzles he calls "Happiness Cubes." His designs include a 3-D version of Duijvestijn's order-21 dissection.

Iwase has a version. (I don't have this.)

In 1978 at a conference at Miami University, Dean Hoffman posed the following problem, which has come to be known as Dean Hoffman's Packing Problem, or the Sugar Lump Puzzle:

Pack 27 cuboids with sides A,B,C into a box of side A+B+C, such that: (1) A,B,C are all not equal, and (2) the smallest of A,B,C must be larger than (A+B+C)/4. There may be voids, but all sides will be flush.

Example dimensions are: 18,20,22 with box 603; or 4,5,6 with box 153 (Cutler).

Cutler says there are 21 solutions, none having symmetries. See Bill Cutler's article Block-Packing Jambalaya.

Several examples have been produced: by John Devost, by Trevor Wood, a cheap monkeypod wood version available at www.gemanigames.co.uk, and a version by Trench Puzzles called The Troublesome Twenty-Seven with mahogany pieces and a flimsy plastic "box." I acquired the latter in an auction from the Ergatoudis collection.

3-D Packing Puzzles with Identical or Similar Pieces


Pack It In - Thinkfun
This is "Conway's Curious Cube" which calls for three 1x1x1 cubes and six 1x2x2 blocks to be packed into a 3x3x3 box. There is only one solution - see this source.

Nine rhombic pieces fit in the tray. This is isomorphic to Conway's Curious Cube.

17 piece packing cube
Another John Conway design. 5 of 1x1x1, 6 of 3x2x2, 6 of 1x2x4. Fit into 5x5x5. The same pattern should show on all sides. Gemani calls this "Made to Measure." I've also seen it as "Shipper's Dilemma."

Conway Box Deluxe
This is a nicer version of the 17-piece Conway cube.

The Meiji Caramel puzzle is a version of Anti-Slide designed by William Strijbos. Pack 15,14,13, or 12 of the 15 1x2x2 pieces into the 4x4x4 box such that none can slide in any direction. There are no solutions using less than 12 pieces. Using 12 pieces there are only three solutions, but using 13 pieces there is only one solution.
This puzzle won 2nd place in the 1994 Hikimi Wooden Puzzle Competition.
Purchased from Torito.

36 piece Packing Puzzle

T Party - B&P

Loyd's Cube - Sam Loyd
An IPP Puzzle from Jerry Slocum

L-Bert Hall
Pack the nine identical pieces into a 3x3x3 cube seated in the box. Each piece is a concave tri-cube with holes and one dowel. This was designed by Ronald Kint-Bruynseels for IPP27, and made by Eric Fuller. The pieces are made from Cocobolo and the box is made from Lacewood.

"The Five Minute Puzzle That Might Take a Little Longer"
Designed by Andy Turner
Entered in the IPP 2009 Design Competition
Made by Eric Fuller, from Oak (box) and Paduak

Wim Zwaan - Octahedron and Tetrahedron
Fit the Wenge tetrahedron into the Baltic Birch plywood Octahedral box. Then get it out again. Since the opening and the tetrahedron are not quite regular, this is more difficult than it might at first seem. Purchased from Wim at IPP28 in Prague.

Mine's Cube of Cubes
Designed by Mineyuki Uyematsu in 2004. Exchanged at IPP24.
14 pieces pack into a 5x5x5 box. 2 solutions.

Mmmm
Pack the four M-shaped pieces into the box and close the lid.
Designed by Hirokazu Iwasawa (Iwahiro).

Mmm
Pack the three M-shaped pieces into the box and close the lid.
Designed by Hirokazu Iwasawa (Iwahiro).

Logs in Box designed by Vesa Timonen
Produced by Hanayama in their "Woody Style" line.
These puzzles are all based on the same design:


Aha Rectangle - Thinkfun; Log Stacker - Elverson; Logs in a Box - B&P; Lox in Box designed by Vesa Timonen
8 pieces, each with a beveled end, to be fit into the tray.


Cherry Cocktail
Pack six pieces - 3 each of 2 kinds - plus the "cherries" into the "glass."
Purchased from Irina Novichkova at IPP28 in Prague.

Thick 'n' Thin No. 7
Purchased from Serhiy Grabarchuk at IPP28 in Prague.

Bermuda Hexagon designed by Bill Cutler in 1992 (using a computer), made by Tom Lensch
12 pieces to be packed into the hexagonal case in 3 layers.
This design was awarded the 3rd prize in the 1992 Hikimi Wooden Puzzle Competition

"Old Hand Cranes 1 Gin"
Eight different blocks to be packed in the wooden Sake cup
designed by Nob Yoshigahara
produced by Hikimi

Cubes in Space, designed and exchanged at IPP32 by Hirokazu Iwasawa (Iwahiro), made by DYLAN-Kobo
An anti-slide challenge.

This is one of Trevor Wood's Teaser Tiles puzzles. Nine tiles, each composed of two slightly different sized layers, with various overhangs. The objective is to fit the pieces flat into the box - i.e. so that all pieces have their two layers parallel to the bottom of the box. Obviously, the particular juxtaposition of piece edges and overhangs will be crucial. Thanks, James!

 
Hoffman's Packing Box, or
The Inequality of the Means Puzzle - produced by Creative Crafthouse
Fit the 27 identical blocks into the frame.

Based on the fact that the arithmetic mean of three values a, b, and c is always greater than or equal to the geometric mean of those values. Symbolically, (a+b+c)/3 >= (abc)^1/3
Starting with the above inequality, cube both sides - we get (a+b+c)^3 / 27 >= abc, then multiply through by 27 to get (a+b+c)^3 >= 27 * abc
This means if we have 27 blocks whose dimensions are axbxc, we should be able to fit them into a cube whose edge length is (a+b+c).

 

3-D Packing Puzzles using a Set of Related Pieces


Nob's Never Ending
Build a cube within the box, from 8 similar angled pieces. The one on the left is a rough handmade version - an auction win. I recognized this in a pic of Matti Linkola's exhibition, and found it on Trevor Wood's site. It is a copy of Nob's Neverending puzzle.
Torito sells a version made by Himiki.

Make Room - variation of Stewart Coffin's #127, by Mr. Puzzle Australia
Craftsman version in fine exotic woods - the box is a waxy wood called Yellow Leichardt.
Four challenges:
  1. Pack all 8 blocks under the closed lid - 30 solns
  2. Pack 8 blocks plus the brass rod under the closed lid - 4 solns
  3. Pack 8 blocks plus the wooden dowel under the closed lid - 1 soln
  4. Shipping config - pack 8 blocks with the dowel through the hole in the lid - 3 solns

This is Tube-It-In by William Strijbos. (Photo from John Rausch's site.)

The Morph
A cube dissected into four clever pieces can morph into three different solids to fill the compartments in the case.
According to Bernhard Schweitzer, who sells a copy, this was designed by Boris A. Kordemsky of Russia.
I believe this was issued by Bits and Pieces quite some time ago, but I am not sure. I found my copy on auction.

Funny Cubes - designed and made by Tom Lensch. Purchased at IPP 29 in SF. Each piece consists of two attached rectangular blocks that can be rotated relative to each other. Fit the blocks into the square tray so that the top pieces also form a square.

Devil's Gate - designed by Ferdinand Lammertink, made by George Miller
Purchased from George during a visit to the Puzzle Palace during IPP 29.
This is a version of the Langford problem. Find lots of info on the Langford problem here.

George Bell's Nine Bed Nightmare assembly puzzle
(May be available from Puzzlewood.de.)
Pieces bot. to top, L to R: A, B, C, D, E, F, G, H, I.
Challenges:
  • 4x4x4 using A,B,D,G,H - 6 solutions.
  • 4x4x5 using B,C,G,H,I or B,C,D,E,G,H or A,B,C,D,H,I or A,B,D,E,G,I - 15 solutions.
  • 4x5x5 using A,B,D,E,G,H,I - 2 solutions.
  • 5x5x5 with all pieces - 1 solution.

Eric Harshbarger's Digits in a Box
Ten size 1x3x5 digits - just pack them into a 5x5x5 box.
The first version, with pieces laser-cut from colored acrylic, was exchanged at G4G9 - I purchased a copy at Eureka.
Popular Playthings now offers a nice mass-produced version.

16 Hexominoes in a Twin Square, designed and made by Marcel Gillen, exchanged at IPP32 by Carlo Gitt

L-omino Cube 4, designed and exchanged at IPP32 by George Bell, made by Ponoko

O'Beirne's Cube, or "Morph 2" - designed by T.H. O'Beirne, made by New Pelikan Workshop, exchanged at IPP32 by Peter Hajek

3-D Packing Puzzles using an Assortment of Dissimilar Pieces


Bunchgrass Packing Puzzle "13/14"
A box with 5 pieces made of spheres - the pieces fit in the box with or without a single sphere piece. They also can form a square-based pyramid. It is called the 13-14 puzzle since with the single sphere there are 14 pieces and 13 without.

For Your Own Sake - Hikimi (Japan)
This puzzle poses the additional challenge of embedding 3 marbles.

Dragon's Eggs - Pentangle
Find a way to pack everything into the box so that the three "eggs" are all concealed.

Slot Machine - Stewart Coffin #185
obtained from Henry Strout
Build a cube within the box, fitting the pieces in through a small slot in the acrylic cover.

18-Piece Mini-Cube-Block Puzzle Set

Four Square
Fit the four dual-layer pieces into the tray.

Back in the Box
A dissection of a cube into various tetrahedra.

Mosaic - IQ Puzzles (Family Games)

Third Degree - Bits and Pieces
Designed in 1995 by Bill Cutler, who calls it the "3-Piece Blockhead." Discontinued.

Stark Raving Cubes / Sneaky Squares / Blockhead / Blockout
I bought mine from ISHI.
Designed in 1983 by and still available from Bill Cutler. Thinkfun's Blockout is a nice, portable, inexpensive copy.
Awarded the Grand Prize at the 1986 Hikimi Wooden Puzzle Competition.

Three Pins
By Jean Claude Constantin.
Fit the six pieces in two layers into the tray, aligning holes so that the three pins can be inserted, each through two pieces.

Pack 6 - Eric Fuller
Entered in the IPP 2003 Design Competition.

Sandwich - Vaclav Obsivac

HCP1 - Vaclav Obsivac

Malaga Box - Philos
By Markus Goetz.

Barrel Puzzle
From Brian Menold at Wood Wonders

Stack the disks to form a cob. This seems to be a copy of the Toyo Glass puzzle "A-Maize-Ing."

Booze Crate

Something Fishy
also in wood - Fisherman's Dilemma, from
Creative Crafthouse

A nice little packing puzzle handmade in the Ukraine. Purchased off eBay.
I'm not sure, but I think this is the same puzzle as shown on www.golovolomki.ru in the Wooden Puzzles section, called "Disobedient Particles" by I.A. Nowitschkowa.

6-piece packing (Krasnoukov?) - from Rick Eason

Dice Box - Sticks

Dice Box - Prisms

Trevor Wood's Cube the Square - unknown craftsman
The 8 pieces form a 4x4x4 or an 8x8x1.

Nob's LL Puzzle - unknown craftsman
Each of the 8 pieces is made from two L tricubes. They pack a 3x4x4 box, made from purpleheart.

Boxed LUV
Stewart Coffin #189

a cheap Asian copy, but functional

Russian 3-piece packing
Obtained from Rick Eason at NYPP 2008. The label is in Cyrillic and I cannot read it!
I'm not sure, but I think this is the same puzzle as shown on www.golovolomki.ru in the Wooden Puzzles section, called "Pythagorean Trousers 2" by I.A. Nowitschkowa.

This is Packman by Gary Foshee. Get all of the elements into the cube so that all of its surfaces are flush. (Photo from John Rausch's site.)

Meiji Apollo
Fit the plastic candy replicas into the box in two layers.
Purchased from Torito

Oskar van Deventer's Two-Piece Packing
From Bernhard Schweitzer at NYPP 2008

Quadron by Naef (1987)
Designed by Jost Hanny

Fragmented Cube - Oskar van Deventer
Pack eight pieces into the box. They can be packed such that faces appear with and without "holes."
Purchased from Oskar at IPP28 in Prague.

Magellan - Philos
Designed by Georg Pfaffinger
12 pieces pack into the 4x4x4 box and leave a 2x2x2 hole in the center. Includes other challenges. Purchased at a puzzle store in Prague.

Caboose, designed, made, and exchanged at IPP32 by Henry Strout

Nine Parts Packing, designed by David Goodman, made and exchanged at IPP32 by Dor Tietz

Six Cushion Shot, designed and made by Wayne Daniel, exchanged at IPP32 by Marti Reis

Tetracubed, designed by Robert Reid and George Miller, made by Wayne Daniel, exchanged at IPP32 by Stan Isaacs

Pack six pieces in the box,
from Japan - purchased at IPP32 (I don't know the name or the designer)

Hikimi "For Your Own Sake" 2-1
A six-piece "squashed" variant of Conway's Curious Cube - contains two voids when solved.

Super Box 2, designed by J.C. Constantin. Pack the six pieces in the tray so none stick out. The dimensions are such that several arrangements seem like they might fit if you just force the pieces a wee bit - but do not! The solution does not require any forcing at all.

P-26 Drehcube, designed by J.C. Constantin. Arrange the six pieces in the tray and form a 3x3x3 cube. Each piece is composed of four unit cubes and a half-cube triangular prism. On five of the six pieces, the triangular unit can be rotated in place to add to the confusion.

IQ Fit from Smart Games.
Designed by Raf Peeters.
I found this at a Barnes & Noble store near me.


Back2Back - designed by Raf Peeters, issued by Smart Games
Place the pieces into the board, using both sides of the board. A polyomino-type packing puzzle, but with an added dimension - the pieces have some units that push through the grid and occupy spots on both sides, and others that occupy a spot on only one side, allowing another piece to overlap on the other side of the board.

I found the IQ Steps in a local Barnes and Noble.
It is another puzzle designed by the incomparable Raf Peeters and issued by Smart Games.



Parcel Post - issued by Jean-Claude Constantin,
purchased from PuzzleMaster.ca
A recent production, in a convenient size (the tray is 100x75x15mm),
of a vintage 18-piece packing puzzle whose designer is unknown.
The U.S. Post Office began domestic parcel post service in 1913.
Vintage copies of this puzzle show a period postal truck on the cover, and say "Parcel Post Puzzle - Trade Mark."
I have been unable to find a reference to the trade mark or any relevant patent.


I found photos online of two instances of the vintage puzzle. In both cases, the cover is quite faded. However, one can make out the "Parcel Post Puzzle" title, and the words "Trade Mark" below it. The truck drawn on the cover is consistent with the style of trucks circa 1913. I also found a photo of a button celebrating the 1938 25th anniversary of the parcel post service. However, the style of truck from 1938 is very different from that depicted on the puzzle cover, so I think the puzzle dates from the earlier period.

3-D Packing Puzzles by Toyo Glass

The Japanese company Toyo Glass issued a series of packing puzzles using glass elements (usually an assortment of plastic pieces which must be packed into a glass container). I have these unless otherwise noted. Much of the Toyo lineup has been re-issued by Beverly (www.be-en.co.jp) - a Japanese vendor. Puzzlemaster.ca carries them.

[A] means the Glass Puzzle Answer Book contains a solution.
[B] means a re-issue is available.


Packed in Tokyo
I got this in Japan.

Java Tea [A]

Packing Peanuts [B]

Shot You

On the Rocks [A]

Pack the Asparagus
Designed by Nob Yoshigahara
Related to Tridiamonds


Pack the Beans [B]
(I don't have this one.)

Pineapple Delight [A] [B]
Related to Pentominoes

Pack the Pudding (or Custard) [B]

Pack the Beer [B]

Pack the Plums [A] [B]

Pack the Peanuts [B]

A-Maize-Ing [A] [B]

Pack the Rice Crackers [A] [B]

Pack the Orange [A]
(I don't have this one.)

Home Alone Husband

Bin Cross [A]

Pentominoes and Other Polyforms

A regular polygon is a closed two-dimensional shape having some number of identical line-segment sides, joined at identical angles. They begin with the equilateral triangle, and proceed with the familiar square, pentagon, and hexagon, then continue with the perhaps less familiar heptagon, octagon, nonagon (or enneagon), etc.

Polyforms (Wikipedia entry) are pieces made by joining multiple copies of a given unit element which is a polygon. In the most straightforward cases, the unit elements are regular polygons and they are joined along full edges. These are also known as animals. The pieces can be distinguished by whether they are convex or non-convex. A piece is convex if you can join any two points inside the figure by a line segment that also lies entirely within the figure. Also, if a piece is distinct from its mirror image, it is chiral, otherwise it is achiral.

Polyforms can also be constructed using three-dimensional unit elements, such as cubes or spheres, and these are referred to as solid polyforms. Solid polyforms made from unit cubes are polycubes. Read about polycubes at The Poly Pages. When solid polyforms are constructed, some of the pieces will have all their unit elements lying in one plane, and others will not. The former are planar pieces, and the latter are non-planar pieces.

Two-dimensional polyform puzzles utilize some set of polyform pieces to create a given two-dimensional shape. Only three regular polygons can be used to tile the plane without holes - equilateral triangles, squares, and hexagons. Naturally, most polyform puzzles have utilized pieces composed of such units, but other polygons can be used. Here are some of the better-known planar polyform types:

The polyominoes start with a single unit, called a monomino. Two units joined along a full edge make a domino; three a tri-omino or tromino, four a tetromino, and five a pentomino. The set of all possible tetrominoes are the shapes used in Tetris. Note that the dominoes referred to here lack the patterns of a conventional set of dominoes, and as a rule, polyomino puzzles do not typically employ pattern constraints other than the occasional checkerboard coloring.


Shapes in a plane may be identical to each other after certain operations are performed:


When enumerating piece sets, it is important to know how to treat each of the operations. There are usually three figures of interest:


Below is a chart of the number of pieces as n grows. Also see Michael Keller's page, Polyomino Enumerations, Joseph Myers' page, and Miroslav Vicher's page. There is no formula known which will give the exact number of all possible pieces given a number of unit elements.

Enumerating Polyforms
Sloan's sequences given for: # free . # 1-sided (holes allowed)
Wolfram links at top show initial pieces; links in table to Wolfram, Ishino's site, etc. show all pieces
n, prefix -iamonds
#A000577
.#A006534
Wolfram
-ominoes
#A000105
.#A000988
Wolfram
-hexes
#A000228
.#A006535
Wolfram
Wikipedia
-aboloes
(-tans)
#A006074
Wolfram
Esser
-cubes
#A038119
.#A000162
Wolfram
Comments
2
d[i]-
1.1
1.1 1.1 3 1.1 Dick Hess designed a puzzle using nine planar tridiamonds.
The Naef Favus puzzle pieces are a set of planar and non-planar solid tri-diamond prisms.
(Labeled dominoes are discussed in the Pattern section.)
3
tr[i]-
1.1 2.2 3.3 4 2.2 The two triominoes consist of one three-in-a-row and one "L" - the L is non-convex.
4
tetr[a]-
3.4 5.7
7.10

14.22
7.8

Tetromino sets: Tenyo BtC #783
See my diagram of polyhexagons up to tetrahexes.
Naef's Hexagon puzzle uses the set of 7 free tetrahexes, made from metal nuts.
The Snowflake puzzle by Stewart Coffin uses the set of three trihexes and seven tetrahexes.
Michael Keller shows some figures and solutions made with the set of tetratans.
The Eternity Delta puzzle is a commercial set of 14 tetratans.
Kadon's Tan Tricks I includes 2 monotans, 3 ditans, and the 14 tetratans.
Jurgen Koeller discusses tetracubes.
The eight tetracubes are named: I O L T N, tower-right, tower-left, and tripod. They can make two boxes: 2x4x4 (1390 solutions) and 2x2x8. A set called Wit's End was produced by Lowe in 1967.
Piet Hein's famous Soma cube uses the six non-convex tetracubes plus the single non-convex tricube.
5
pent[a]-
4.6
12.18
22.33


30.56 23.29
12 planar
17 non-p

Ishino's page on pentiamonds.
Peri Spiele (Austria) makes a set of 19 n-iamond pieces packed into a Star-of-David tray. The set includes two tetriamonds, seven pentiamonds (all 4 possible + dups), six hexiamonds, three heptiamonds, and one octiamond.
The 12 planar pentomino pieces are named by convention after the letters they resemble:
F I L N P T U V W X Y Z.
There are too many commercial pentomino sets to mention.
Ishino's page on pentahexes.
Commercial sets of pentahexes: Tenyo BtC #22, Hi-Q Fusion, Hi-Q Confusion
Kadon's Tan Tricks II includes the set of 30 pentatans.
There are 29 pentacubes - 12 planar (corresponding to and named like the pentominoes) and 17 non-planar.
Stewart Coffin on solid pentominoesStewart Coffin's Unhappy Childhood puzzle
Kadon's page naming the planar pentacubesKadon's page naming the non-planar pentacubes
6
hex[a]-
12.19
35.60

82.147 107 112.166 Ishino's page on hexiamonds.
Hexiamond sets: Tenyo BtC #6
Hexomino sets: Tenyo BtC #600, Spear's Multipuzzle
George Miller sold a set of 82 hexahexes.
Kadon's Tan Tricks III includes the set of 107 hextans.
Kadon sells a set of 166 hexacubes.
Livio Zucca's Sexehexes (not polyforms)
7
hept[a]-
sept[a]-
24.43
108.196 333.620 318 607.1023 Ishino's page on heptiamonds.
Heptiamond sets: Tenyo BtC #24
Kadon sells a set of 108 heptominoes.
Peter Esser's page of the 108 heptominoes.
8
oct[a]-
66.120 369.704 1448.2821 1116 3811.6922 Kadon sells a set of 66 octiamonds.
Ed Pegg Jr.'s page on octiamonds.
Kadon sells a set of 369 octominoes.
9
non[a]-
enne[a]-
160.307 1285
.2500
6572
.12942
3743 25413
.48311
George Miller sold a set of 160 noniamonds.
10
dec[a]-
448.866 4655
.9189
30490
.60639
13240 178083
.346543
 
11
endec[a]-
1186
.2336
17073
.33896
143552
.286190
46476 1,279,537
.2,522,522
 
12
dodec[a]-
3334
.6588
63600
.126759
683101
.1364621
  9,371,094
.18,598,427
 

Perhaps the best known variety of polyominoes are the Pentominoes. Hexominoes and Heptiamonds are also used in puzzles, but the number of pieces quickly becomes unwieldy as one goes up from there.


There are many websites devoted to polyforms and polyominoes in particular.

Basic pentomino challenges include fitting the pieces into a rectangle, or a square with some holes. You can also form large models of each pentomino!

If you become bored with the basic pentomino puzzles, several people have devised more interesting challenges...

Often Pentominos are presented as a packing puzzle, but they are very versatile. If they are made from unit cubes, they can be arranged either flat or in 3 dimensions. However, the 3-d constructions do not really interlock due to the limited size and convolution of the pieces.


Concept 5

Yasumi

University Games
Pentomino Set

Logika
 
Kohner Hexed (thick and thin versions, and alternate cover)

Choc-a-Bloc - from Kidult
A Pentomino set - the 12 pieces resemble
chocolate bar pieces and can be flipped over.
Presented in a clear 6x10 case.
Pentomino sets made into games:

Quintillions, a nice Pentominoes set, by Kadon.
This product launched Kadon in 1979.

ZahlenLabyrinth - Logika

Camelot (castle pieces
on top of flat pentominos -
arrange the pieces to build the castle)

Springbok Pentominoes

The 12 planar pentominoes can be fit into various rectangles: The 12 planar solid pentacubes can be packed into various boxes:
  • 2x3x10 - 12 solutions
  • 2x5x6 - 264 solutions
  • 3x4x5 - 3940 solutions

See Chapter 3 in Stewart Coffin's The Puzzling World of Polyhedral Dissections.

Here is one of the 3x4x5 solutions, in case you need to put your set back in its box...

I I I I I   X F N L L   Y Y Y Y T
X V V V T   X F N L T   X F Y Z T
U F N V P   X F N L P   U Z Z Z T
U W N V P   U W W L P   U Z W W P


Checkerbox - Bill Cutler
12 checkered pentominoes pack into a 3x4x5 box


 
Wit's End by Lowe from 1967 is a set of tetracubes. The instruction sheet gives several construction problems.

The Spear's Multipuzzle is a plastic set of hexominoes. It includes all 35 "free" hexominoes and duplicates of 7 of them. The pieces are essentially 2D - they are not built from unit cubes and cannot be built into 3D structures. The set comes with a 6x10 tray and a booklet of problems specifying subsets of pieces to be fit into the tray.

The Ten Yen puzzle, published in 1950 by the Multiple Products Corp. of NY, includes a monomino, domino, both trominoes, and 3 each of the tetrominoes and pentominoes. Kadon offers one. Pieces in three colors. One challenge is to create identical shapes from the sets of three different colored pieces.

A gift from Brett of three "Meiji Chocolate" plastic Polyomino puzzles by Hanayama - Milk (12 pentominoes), Black (11 hexominoes), and White (8 pieces) - find them at Kinokuniya.

3-4-5 iamonds - designed by Koshi Arai - IPP30
5 sets of the four pentiamonds. The 20 tiles will pack into both sides of the tray - the large triangle, and the double-layer diamond.


Tenyo made several polyomino puzzles in their "Beat the Computer" series.
Almost every set has been produced in a variety of color combinations. Several similar sets have been offered by other vendors as well.
Pascal Huybers has a webpage showing most of them.

#0

#0
When circles are arranged into a hexagonal grid, there will be six equally spaced triangular interstices around each circle. The #0 has 13 pieces - including all of the 12 ways a circle can be augmented with from one to six triangular interstices, plus an extra "single." The pieces can be flipped over.

Kwazy Quilt by Kohner is equivalent to Beat the Computer #0. I have two Kohner versions - thick pieces (shown on the left) and thin pieces. Another version (I don't have) is called Hi-Q Euclid by Gabriel.

From King's, a smaller version of "Kwazy Quilt."

In BurrTools, you can model the Kwazy Quilt pieces on a triangular grid as hexagons, each having from one to six triangular points. BurrTools reports 3594 solutions, but I suspect this includes some symmetrical positions. I found the particular solution shown by hand.

I saw this variation from "Peri" on someone's web site - I do not have this puzzle. It uses 19 pieces but not full sets.

This "Wisdom Puzzle" includes only seven of the Kwazy-Quilt-type pieces. Select one and place it in the upper left hand "Begin" position. Then try to fit in the rest. 120 combinations in total.
#5 and #8

#5 (left) and #8
The 12 pentominoes - #5 packs them in a 6x10 tray. #8 packs them in an 8x8 square tray with a 2x2 hole.
I don't have these, but see above for several other pentomino sets.
#6

#6 is a set of Hexiamonds - each piece is formed from six equilateral triangles - all packed in a truncated triangular tray. There is also a version in an alternative tray. I don't have either Tenyo #6, but I have several versions from other vendors: The green set is called "Triconometric Puzzle #24" and is made by Lucky. Peter Pan and Java made sets of Hexiamonds, and I found a misc. yellow set, too.
#22

#22
A set of the 22 pentahexagons.
I also have Hi-Q Fusion and Hi-Q Confusion.
#24

#24
The 24 heptiamonds. I have sets by Lucky (the orange one is called "Triconometric Puzzle #4503") and a black one called "Heptamond" made by Tenyo and purchased from Torito. I don't have the blue Tenyo set.
#600

#600
This is a set of the 35 hexominoes.
#783

#783
783 comprises
two sets of the tetrominoes. I have two examples.


I obtained a group of assorted polyform tray packing puzzles - some of which are duplicates of puzzles shown elsewhere, others of which are tangram and sliding piece puzzles.

Peri Spiele (Austria) makes a set of 19 n-iamond pieces packed into a Star-of-David tray. The set includes two tetriamonds, seven pentiamonds (all 4 possible + dups), six hexiamonds, three heptiamonds, and one octiamond. I also found a set that says "Puzzle" instead of "Peri" in the black circle on the box.


A one-million pound prize was offered for the solution of the Eternity Puzzle.
I didn't win. The puzzle comprises 209 pieces called 12-polydrafters.

For more info on the Eternity series, take a look at

The Eternity Delta puzzle was billed as a warm-up to the full Eternity. It uses the set of 14 tetratans.

Here are some interesting sites discussing polytans:


This is the Eternity Meteor puzzle. It uses a set of ten penta-hexagons.

Last but not least, the Eternity Heart.


More puzzles using poly-hexagons...


I believe this is "Hextra" from Robert Longstaff Workshops. It uses a set of septa-hexagons. This is a gift from Carol Monica, the proprietress of one of the best puzzle shops around - the Games People Play shop in Cambridge, Mass.

The Snowflake puzzle was designed by Stewart Coffin (#3), and this version made of foam was offered by Binary Arts in 1993. It includes two sets of 3 tri-hexagons and 7 tetra-hexagons, a tray with two levels, and a booklet of challenges.

Here is an unnamed but colorful set of tetra-hexes in a clear case.

The "Hexagon Sense-A-Gone" is one in a series of Brain Drain puzzles from Mattel. It employs a set of 3 tri-hexagons and 7 tetra-hexagons. The pieces cannot be flipped, and only one of each of the pairs of mirror images is used. The pieces are prettily colored and suggest 3-dimensional cubes, but the instructions do not indicate any edge-matching constraint. Assemble them / Pack them in the tray.

This is a diagram of the family of poly-hexagon pieces up to tetra-hexagons:

This is as good a place as any to show the six Mattel Brain Drain puzzles from 1969 (that I know of)...


Hexagon Sense-A-Gone
Assembly

Profound Round
Circle Dissection

Mangle Quadrangle
Edge Matching

Checkle Heckle
Checkerboard Dissection

Block Shock
Edge Matching

Square Where
Packing
Equivalent to the Pressman Think Square puzzle.


Other planar polyform puzzles:


Kadon Rombix

Galt Puzzle Blocks

TriPentaHexagon - George Miller

Penguins on Ice - SmartGames - Raf Peeters
Raf has created another great multi-challenge puzzle, this time based on Pentominoes.
But instead of requiring 12 pieces, Penguins on Ice uses only five - each of which can assume multiple shapes!
Part of each piece is able to slide relative to the rest of the piece, and so each piece can be transformed among multiple shapes.
Each piece also has a penguin attached at one spot. A series of challenges designate the required placement of a subset of penguins.
Thanks, Raf!

IQ Twist from SmartGames, by Raf Peeters
A series of challenges, each of which stipulates the placement of specific colored pegs in the board. You must then fit in the 8 pieces so that each peg matches a hole in a piece and the color of that piece.
A cross between an assembly/packing puzzle and a pattern puzzle, but I feel it's more of an assembly challenge at heart.

Polycube and Other Solid Polyform Puzzles



Piet Hein's Soma Cube is the classic example of the polycube puzzle. The Soma Cube uses the six non-convex tetracubes plus the single non-convex tricube.

Pictured above are: a pair of plastic Soma cubes from Parker Brothers; a wooden Soma on an aluminum base - the wood is beautiful - dark and striated - I believe it's Rosewood; the green felt base is stamped "Produced in Denmark" though some of the text is damaged; a Soma Cube I made from Lego; Skor-Mor's Fascinating Cube.

The Balanced Soma is an assembly such that the pieces remain together when balanced on a single cube placed at the center of the bottom face. At least six such constructions exist.

The eight pieces of this Baumeisterspiel ("Master Builder") set from Logika include the Soma pieces, plus a 1x1x3.
I also have a "mini" version with a handy cover.

Rhoma is like Soma, but with rhombic pieces. I have a large and a small Rhoma.

The Illusions from Magnif is similar to Rhoma.
  • Mellow Yellow - Not Too Hard (1142 solns)
  • OK Orange - Somewhat Hard (30 solns)
  • Mean Green - Fairly Hard (16 solns)
  • Rough Red - Really Hard (1 soln)
  • Baffling Blue - Extremely Hard (8 solns)
  • Perfect White - Incredibly Hard (1 soln)
  • There was also a brown, identical to the Red.
The Impuzzables line of 3x3x3 polycube puzzles were some of the earliest introduction I had to mechanical puzzles. I was able to purchase more on a vacation trip to the Great Smoky Mountains.

Here is a link showing the pieces of the Impuzzables.

The Impuzzables are also described on p. 3^3-13 of Kevin Holmes' and Rik van Grol's book "A Compendium of Cube-Assembly Puzzles using Polycube Shapes," which also discloses the number of solutions for each.


Bill Cutler's Splitting Headache
yields a nice A-Ha moment when one
solves it systematically.
I think I bought this at Games of Berkeley many years ago.
Discussed on Peter Kaldeway's site.

Stewart Coffin's Half Hour Cube (#29)
see the pieces at Puzzle Will Be Played... ; also see Chapter 3 in Puzzling World of Polyhedral Dissections (scroll down to Fig. 53)

The TetraCube
Purchased from Wingstoys (defunct). Cheap Monkeypod wood. 13 pieces make a 4x4x4. One "L" tetracube, plus 12 pentacubes: 6 planar: F, L, P, T, W, Y, and 6 non-planar, 3 pairs of mirror images: (using Kadon's naming system) L1 and J1, L2 and J2, and L4 and J4.

The Bedlam Cube
Wikipedia entry

Bedlam Treasure Chest
Gift from Brett. Thanks!

The Pedestal Problem has cubies joined at an offset, and must be assembled inside fenceposts

The craftsman Scott T. Peterson of the state of Washington made this beautiful version of Stewart Coffin's Unhappy Childhood (#41) puzzle for me.
Of the 17 non-planar solid pentominoes, 12 lack an axis of symmetry. Eliminate the two that fit into a 2x2x2 box to arrive at the ten pieces of this puzzle. Those ten pieces pack into a 2x5x5 box in 19,264 ways, and can be checkered in 512 ways. Only one of those possible checkerings has a unique solution (one other has no solution and the rest have multiple solutions) - this is the checkering for the Unhappy Childhood.

I also obtained a version made by Jerry McFarland, called "Coffin's Cuboids." The diagram will help you pack your set into a 2x5x5 box, though it is not a solution since the arrangement shown won't make a proper checkerboard pattern.


Cube from Melissa & Doug - the same set of planar pieces as the classic Diabolical Cube, which appears in Hoffmann's 1893 Puzzles Old and New. Also see Kevin Holmes' Compendium, page 3^3-3.

Metropolis

Rubik's Bricks

Naef Gemini
Designed by Toshiaki Betsumiya.
Ten pieces - all ways of joining two 1x2x2 blocks. Make a 4x4x5 block - 25 solutions. See the Gemini pieces here.

Stewart Coffin's Patio Block (#82)
made by IP.
Eight pieces form a 4x4x4. Same pieces as Gemini, but omits the 2x2x2 and the 1x2x4, and substitutes a duplicate for one other piece, since the original eight remaining cannot form a 4x4x4.

Naef Campanile
Designed by Manfred Zipfel and Cordula von Tettau in 1979.
See the Campanile pieces here.

Professor Brain's Tower Puzzle
10 pieces, different from Campanile.

Here is a puzzle using pieces made from unit spheres - the pieces stack inside a cage. It is called "Cerebrum."

Flogik.de Skyscraper
This is almost identical to Naef's Campanile (but made with much less quality). In the Skyscraper, piece 'B' has an extra cubie sticking up at the junction.

Tourelle - designed by Yavuz Demirhan, made by Brian Menold
from Sheoak and Cocobolo
Fit seven pieces into the frame - 1 solution.
Similar idea to Campanile but there are no planar pieces and the entryway restricts piece orientations.

Double Cross (without the tray) (discontinued) from William Waite. Fit the 6 pieces together in 2 layers of 3. I think I actually prefer it without the tray - the pieces mate tightly and seem like they would be difficult to manipulate if they were in a tray.

Naef Escalon
Designed by Jost Hanny.

Tetris Cube
Designed by Matt Campbell, produced in 2007 by Imagination Games and tetris.com. 9839 solutions - confirmed by BurrTools. This is the small-sized cube.

Eclecticube - Kevin Holmes

Double Take - Mag Nif 2003
Eight pieces form a 4x4x4 cube or an 8x8 square.

Albertuv #4
The eight octacubes form a 4x4x4 cube or an 8x8x1. Purchased at a puzzle store in Prague.

Albertuv #8
The eight octacubes form a 4x4x4 cube or an 8x8x1. Purchased at a puzzle store in Prague.

KeshIQ erasers
mfd by Seed Co. in Vietnam. Purchased from Eureka

Dollar Tree Hexagon
Equiv. to Naef Favus at a fraction of the cost! (Favus was designed by Toshiaki Betsumiya.)

Japanese hexagon
An Asian version of the Hexagon/Favus.

Werkstattwürfel 1
Designed by Bernhard Schweitzer

The Question Mark Puzzle from Pentangle - six pieces form a cube in two ways, and also fit into the 3x6 box to form a question mark shape. - This is equivalent to the Steinhaus (aka Mikusinski's) Cube.

Cube Conundrum from House of Marbles
Purchased at the Vermont Toy Museum in Quechee Gorge Village.

Rubik's Puzzle - MegaHouse 2010
Nine planar polycube pieces, stickered using standard Rubik's colors; also a clear cubic container, and instruction sheet (in Japanese).
Includes an "official" 1x1x1. Also 2 1x1x2, 3 tricubes, 2 tetracubes, and 1 pentacube (the 'F').
The pieces can be assembled into a 3x3x3 where the six faces are colored as a standard Rubik's Cube.

4 Uni Cubes - idea by Marcel Gillen, Program by Georges Phillippe
IPP18 (Tokyo) exchange puzzle from Luc De Smet
Includes 7 plastic polycube pieces - the O, L, and T tetracubes, and four pentacubes - two mirror-image pairs N1 and N2, and S1 and S2. Comes packed in the box in a 2x4x4 arrangement. Five challenges - you can remove each of the four pentacube pieces in turn and with the remaining six pieces make a 3x3x3 cube; also, find an alternative to the 2x4x4 solid.

A 7-piece cube by Galley Games.

There are several interesting polycube puzzles I do not have:

Puzzle Pyramids

Assembly puzzles need not be made just from unit cubes combined in pieces to build a yet larger cube - spheres are another common building block, as are building blocks derived from tetrahedrons. The "pyramid" - or more correctly, the tetrahedron - has proven to be a popular shape for assembly puzzles. Vendors seem to like to capitalize on the mythos and mystery associated with the famous Egyptian pyramids, despite the fact that most of their puzzles are actually tetrahedral in shape rather than pyramidal. The two diagrams to the right show how each shape looks when "unfolded" - you can see that though the pyramid (on the left) and the tetrahedron (on the right) both have triangular sides, the pyramid has a square base while the tetrahedron has a triangular base equal in size and shape to its sides.

Simple Pyramids

Here are several "pyramid" puzzles that while each posessing only a few pieces, nonetheless can prove to be quite puzzling!

This is the classic two-piece tetrahedron patented by Edward T. Johnson in 1940 (U.S. Patent 2216915).
It was popularized in the following decade when a small plastic version became available from FUN Inc. of Chicago,
starting in 1956, and has been produced ever since.
The classic 2-piece pyramid has to be one of the most simple yet elegant puzzles devised.
Once you've solved it, it gets old, but it is always fun
to watch a newbie's first encounter with it!
Many other examples have appeared - here are a few I have...


Fire - designed by George Hart
A two-piece dissection of a tetrahedron using a helical cut - this large but loose-fitting 3D printed example served for two months as a hands-on exhibit in a gallery at Stony Brook University.
A gift from George - thanks!
From the two-piece, things escalate...

Here are two examples of an interesting tetrahedron - which, when assembled, will have a void inside. It has three identical (and very pointy!) pieces.


This three-piece tetrahedron was originally offered by Wayne Daniel at Interlocking Puzzles back in 2002.
Dave Janelle at Creative Crafthouse now offers one called the Triangulator. (I don't have this.)

This is Tetra from Design Science Toys (defunct). A 4 piece pyramid.

The four-piece Rosie's Puzzle (No. P25) issued by Drueke,
is a derivative of the classic two-piece.
Adam's Pyramid Puzzle is the same (I don't have it).

The Four Piece Pyramid Puzzle issued by Binary Arts
is a novel dissection of the tetrahedron into four equal shapes.
Thinkfun continues to offer this puzzle, in their "Aha" line.

This four-piece tetrahedron called Tetra Teaser issued by Stokes Publishing Co.
is a yet a different dissection of the tetrahedron into four equal shapes.
Astute puzzlers will recognize a relationship to the original classic two-piece.
Pentangle offers a wooden version they call simply "Pyramid."

King Tut's Pyramid from DanleyQuest.
Same pieces as Tetra Teaser, but
each piece has different symbols on its faces.
An additional goal is to ensure that each of the three
visible faces of the pyramid will have three specific symbols
that signify a certain phrase.

The Pyramidal Pile or Setting Hen suggested by Stewart Coffin, made by Brian Menold at Wood Wonders, from Holly and East Indian Rosewood.
The units are truncated rhombic dodecahedra. See Stewart's discussion of puzzles made from rhombic dodecahedra .

Ball Pyramids

A popular style of pyramid puzzle comprises pieces composed of tangentially joined spheres - these are colloquially known as Ball Pyramids. Piles of unit spheres touching in regular arrangements can be constructed in different ways. In each case, the centers of the spheres will occupy points in a three-dimensional grid known as a lattice. The French physicist Auguste Bravais, who lived in the early nineteenth century (1811-1863), identified fourteen unique three-dimensional lattice types, now known as Bravais Lattices in his honor. The angles that can occur between adjacent spheres within pieces will be dictated by the lattice. Ball Pyramid puzzles employ a face centered cubic lattice for both tetrahedral and pyramidal piles.

George Bell has written several articles about polysphere puzzles, published in the Cubism For Fun (CFF) journal.

George offers several polysphere puzzles for sale at his Shapeways Shop "Poly Puzzles."


A [square] pyramidal pile.
Each layer i, starting at the pinnacle, will have i2 balls in it. The entire pile will have (i * (i+1) * (2i+1))/6 balls in it - Thomas Harriot (1560-1621) seems to have been the first to figure out this equation. Here are the numbers of balls (with the total for the stack thus far shown in parens):
1 (1), 4 (5), 9 (14), 16 (30), 25 (55), 36 (91)...

Note that the smallest pile which can be formed into both a square pyramid, and a flat square, has 24 layers and contains 4,900 balls (that can make a 70x70 flat square)! In 1918, G. N. Watson proved that there are no other solutions.

See the diagram below - the first layer of spheres is laid in a hexagonal arrangement. The second layer goes into depressions formed by the first layer, but will only fit in one of two mutually exclusive subsets of the depressions. Once we have placed the second layer, we now have a choice as to where to place the third layer.

The lower-left-hand image shows an arrangement where the 3rd layer goes into depressions directly above spheres in the first layer. The lower-right-hand image shows an arrangement where the 3rd layer goes into depressions above holes in the first layer. The former is known as hexagonal close packing, and the latter is known as face centered cubic packing. The face centered arrangement is used for Ball Pyramids and Tetrahedrons.


A tetrahedral pile. Each layer i, starting at the pinnacle, will have (i * (i+1))/2 balls in it (these are called the triangular numbers). A tetrahedron with i layers will have (i * (i+1) * (i+2))/6 balls in it (these are called the tetrahedral numbers - here are the numbers of balls (with the total for the stack thus far shown in parens):
1 (1), 3 (4), 6 (10), 10 (20), 15 (35)...

Mag-nif and others have issued a classic ball pyramid (tetrahedron), comprising four pieces each composed of joined spheres. Mag-nif calls their version Tut's Tomb.
The German company Pussycat makes a diminutive equivalent version.

Variations on the Tut's Tomb design (where the basic four pieces have been further divided) have appeared in plastic, metal, and wood...



A metal version from Bits & Pieces. The pieces are 2x 1x4, 4x 1x3.
The House of Marbles Pyramid Puzzle is a wooden version of this (I don't have).

A larger derivative called The Lost Game of the Pharaohs also has six pieces: 2x 1x6, 2x 2x5, 2x 3x4. (Pharaoh sculpture not included!)
The Pyramystery by noted Danish designer Piet Hein (1905-1996) is one of the better-known ball pyramid puzzles. It consists of six planar pieces.


I don't have the wooden version issued by Skode, but I do have the plastic version of Pyramystery, by Hubley.
I also have a wooden copy.


The Cannon Ball Puzzle, issued by Skor-Mor in 1973.
Seven planar pieces of five balls each can be assembled into a side-5 pyramid, in addition to other shapes.
Despite the credit to "John Bird, inventor" on the box, this was invented by Michael Reilly.
You can read an interview with Michael at Eric Shamblen's PuzzleMonster website.
Michael attempted to produce a remake via a Kickstarter project, but it didn't get funded.
However, you can find a remake at Creative Crafthouse.
Reilly is also the inventor of the Oops and Oops Again
ball pyramid puzzles, as well as the game Archieball.

Stan's (Len's) Tetrahedron a 3D print from George Bell
originally thought to have been designed by Stan Isaacs, now known to be a Len Gordon design.
These four pieces do interlock and the tetrahedron snaps and holds together, but I have listed it here with other ball tetrahedron puzzles.

Interlocking Tetrahedron a 3D print designed by George Bell
These four pieces do interlock and the tetrahedron snaps and holds together, but I have listed it here with other ball tetrahedron puzzles.

The three Gordon Brothers pyramids are some of my favorites -
the smaller Perplexing Pyramid is doable by hand,
but I wrote a computer program to solve the Giant Pyramid.
The Big Pyramid has a square base.
You can purchase the Giant and Perplexing, as well as a set called "Warp-30,"
from Kate Jones at Kadon.
I depict some solutions at right.

See another solution to the Giant Pyramid, on Richard Whiting's site.

Perplexing Pyramid
     3      OO = 1  OOO = 2   OOOO = 3
                                      
      4                               
    3  4    OO = 4  OOO = 5   OOO = 6 
             O        O        O      
      5                               
    4  5                              
  3  6  5                             
                                      
      1                               
    5  1                              
  6  6  6                             
3  2  2  2                            
Giant Pyramid
     5        L = 1,2,3,4
              C = 5      
      5       S = 6      
    3  3      P = 7      
              I = 8      
       7      J = 9      
     3  5                
   2  2  5               
                         
       7                 
     3  1                
   2  6  1               
 4  6  9  1              
                         
        7                
      7  6               
    2  6  8              
  4  4  4  8             
9  9  9  1  8            
Big Pyramid
        1         OOOO = 1    
                                   
       2  1        OO         
     8  2           OO = 2,3,4
                              
     2  4  1       OO         
    4  2  5         O = 5,6   
   8  5  5                    
                   OOO = 7,8  
   6  6  3  1       O         
  6  4  3  3                  
 4  8  7  3                   
8  7  7  7                    

Perplexing Pyramids - Gordon Bros. of Fair Oaks CA (c) 1976 item no. 154
Six pieces, equal to the Perplexing Pyramid from Gordon Bros. - but this package includes an additional challenge on the back.
Use the six pieces to make what George Bell calls the "Roof" shape, with a 3x4 base.
A timely find, as in the July 2014 issue #94 of Cubism for Fun, George Bell coincidentally has an article entitled
Pyradox: A Pyramid Packing Paradox in which he analyzes sets of pieces that can form both a tetrahedron and a roof.


Here is a 4-piece puzzle called "Der Fluch des Pharao" (Curse of the Pharaoh) by Markus Goetz, made by Philos and purchased from Funagain Games. The pieces actually do interlock but I still categorize this as an assembly rather than an interlocking puzzle.

Cubikon Ball Puzzle
The pieces of the Ball Puzzle from Cubikon are all planar and have spheres joined at 90-degree angles. Contrast with the pieces of Fantastic Island which employ 60-degree joints. Fit the pieces in the tray, then use subsets of them to make pyramids.

Kanoodle - SmartGames
Fit the pieces in the tray, then use subsets of them to make pyramids.

Puzzle in a Puzzle Box, designed, made, and exchanged at IPP32 by Thomas Beutner

Kolossal Pyramid from Kadon
Designed by Len Gordon; 12 pieces (8 of one and 4 of another). 47 solutions.
I don't have this.

Two-piece Ball Tetrahedron - a 3D print from George Bell
These two pieces do interlock and the tetrahedron snaps and holds together, but I have listed it here with other ball tetrahedron puzzles.

Complex Pyramids

These more complex pyramidal puzzles are all composed of several pieces.


The Bermuda Triangle, designed by Adrian Fisher and
issued by Pentangle, is a five piece wooden tetrahedron.
The pieces do not interlock, and there will be voids inside the completed puzzle.
(Purchased at Cleverwood for $19.)

The Tempil puzzle issued by Dalloz
is a copy of the Bermuda Triangle.
Purchased in auction from the John Ergatoudis collection.

Pyrra was issued by Design Science Toys (defunct). It has 15 pieces and 3 distinct solutions.

This is a ten-piece pyramid. No name or manufacturer info on the box, other than "Mindgame." Purchased at New England Hobby. There are at least two distinct solutions, since I found one by hand that is different from the supplied solution.
The pieces are composed from two logical units - a square-based pyramid, and a tetrahedron (slightly stretched). There are a maximum of two tetrahedrons and 3 pyramids per piece.

The "Bamboo" Pyramid has the same 10 pieces as the Mindgame pyramid shown above. The 10-piece has been offered under several brands.
I don't have these.

Gizeh by Siebenstein Spiele - 8 pieces.
I don't have this.

This is a 9 piece pyramid. I don't have this. Creative Crafthouse offers a version.

Pieces of the 9-piece design can be combined into fewer pieces - above is a 5-piece pyramid.
The 5-piece has also been offered with an attached cord - I have seen it called Khufu's Pyramid offered by Siam Mandalay.
Each apex-forming piece has been combined with its adjoining tetrahedron, and pieces in the base are also simplified -
two of the square-based pyramids have been combined with a tetrahedron to form the largest piece in the base.
I don't have these.

Here are the Lupus, No. 5156 from Philos, in which
the base has been further simplified to only two pieces, making a 4-piece pyramid, and the similar Choups from Dalloz. I don't have these.

Goki pyramid
A five piece tetrahedron with irregular pieces.
I don't have this.

I found this Pyramid Puzzle by Galley Games in a shop in St. Augustine Florida.
It is the same as the Goki Pyramid.

Blue RD Tetrahedron - advertising promo

An Eraser Pyramid
I don't have this.

This is a 14 piece pyramid. Creative Crafthouse offers a version.
I have seen it in various woods and different trays, also a version with a triangular collar piece that I suppose helps hold the assembly together.
This has also appeared as the Luxor puzzle.
I don't have this.

I got the PyrPlex from Andy Snowie.

Philos offers a copy of Snowie's PyrPlex they call Gizeh (not to be confused with the Siebenstein Spiele version) - I don't have it.

Dollar Tree Pyramid
Equivalent to the Pyrra.

This is the Pyramuddle. It seems to have been offered by Duncan Law circa 2011. I don't have this.
The Step Pyramid of Djoser at Saqqara was the first pyramid the Egyptians built. It was constructed about 4,700 years ago during the 27th century BC and planned by the architect Imhotep, vizier of Pharaoh Djoser.

Its layers have a stair-step arrangement rather than the smooth sides of later pyramids such as the Great Pyramid at Giza.

There are several puzzles in the form of Step Pyramids.


This is a Step Pyramid from Philos,
designed by Ferdinand Lammertink,
having 10 pieces.

Here is another step pyramid, from Germany.
It is much smaller than the Philos,
and made of plastic rather than wood.
It uses 7 pieces.

Here is a 6 piece Step Pyramid. I have seen this called the Block Pyramid and the Pagoda Pyramid.
I don't have this.

I have seen this called the Mayan Pyramid. It is similar to the 6 piece shown at left, but has 8 different pieces.
I don't have this.

This a 20 piece step pyramid. It has one solution. Lee Valley carries The Robert Dalloz version of it. Creative Crafthouse offers a version, too. I don't have this.

The Wood and Middleton Puzzle Pyramid or the Tower of Nottingham Task

I found the following interesting information at PsychoHawks

Wood and Middleton (1975) studied the influence of instruction with their experiment. They provided 3-4 year olds with a puzzle which was beyond their comprehension on their own. The mother then provided different levels of assistance for the child:

  • L1 – General verbal instruction (“Very good! Now try that again.”)
  • L2 – Specific verbal instruction (“Get four big blocks”)
  • L3 – Mother indicates material (“You need this block here”)
  • L4 – Mother provides material and prepares it for assembly
  • L5 – Mother demonstrates the operation
After the session, the child was assessed on whether they could construct the pyramid on their own. Results showed that when children were given varied support from mothers (low levels of support when the child was doing well, and high levels when the child struggled) they were able to construct the pyramid on their own. However, when the mother consistently provided the same support, they seemed to make the child conclude the activity was beyond their comprehension and the child soon lost interest in constructing the pyramid. This shows the importance of providing the correct level of scaffolding when teaching a learner.

See Jones, Ritter, Wood 2000 (PDF)
Also see Jones & Ritter 1998 (PDF)

This is a Pyrix puzzle. Assemble a tetrahedron such that each face is a uniform color, constrained by the fixed threading of the pieces. U.S. Patent 5108100 - Essebaggers 1992

From the same maker as Pyrix, Pyram consists of an octahedron and four smaller tetrahedrons, each having various patterns on their faces. Build a tetrahedron satisfying a pattern constraint.

The Pyrus Puzzle completes the three offered by Enpros. Like Pyram, an octahedron and four tetrahedrons. Build a larger tetrahedron having each of the four colors appear on every side.
The following puzzles, while advertised as "pyramids," are of course triangular prisms.

The 11 piece Beehive Pyramid.
I don't have this.

The 3 piece Pyramid.
I don't have this.

 

Dissection Puzzles

There are various styles of dissection puzzle, but all of them involve some figure which has been cut up, or "dissected." The objective is usually to re-assemble the figure. Sometimes the pieces of a dissection are contrived such that an alternative figure can be assembled, too. In some cases, it is even possible to "hinge" the pieces to each other so that both forms can be assembled. See this link at Wolfram for more info on dissections.

Dissected Squares


The Tangram puzzle is a venerable classic where the real objective is to form various silhouettes from the given pieces. However, this version from Melissa & Doug is presented as a straightforward square-dissection and tray-packing problem.

The Magic Square
Make a square from the four identical pieces. According to Frederickson (p.30), this was designed in 1873 by Henri Perigal, who was a London stockbroker and amateur mathematician (1801-1899).

Square Up
Make a square from the four identical wooden pieces. The pieces come arranged with a small square hole in the center - your task is to find a way to make a square containing no hole.

Double Square - Thinkfun
This is another fairly well-known design - form a square from 4 pieces, then add a fifth piece (a small square) to form another larger square.
This design dates back at least as far as the 1934 Johnson Smith catalogue.

The St. Charles Milk Puzzle
Seven pieces form a square. Discussed in Slocum and Botermans' The Book of Ingenious and Diabolical Puzzles on p.12.

Dickinson's Witch Hazel
A vintage advertising promo.

The Elusive Square Puzzle - TSL
Twelve pieces, whose collective area is 32 unit squares. What does that tell you about the solution?
The Pythagorean Puzzle
Originally sold in London in the 1840s
IPP30 exchange from James Dalgety
Use the six pieces to prove the Pythagorean Theorem:
For a right triangle, the sums of the squares on the sides equals the square on the hypotenuse.


Snider's Diamond Puzzle
The 10 pieces form a square.
Discussed in Slocum and Botermans New Book of Puzzles on p.14.
This design has been around for a while and has been called the Egyptian Puzzle.
Assemble a square from the 10 pieces that result from 5 smaller squares each sliced on a bias from a corner to the midpoint of an opposite side.
It is discussed on p.19 of Slocum and Botermans' "Puzzles Old & New."
See U.S. Patent 907203 - Walker 1908

The Horse Blanket Puzzle was used as advertising for blankets made by Wm. Ayres & Sons of Philadelphia.

This twleve-piece version was used to advertise Devoe Paint. Note the kite-shaped pieces resulting from a couple of squares being doubly-sliced.

This cardboard version from 1943 is called "Bombing Mystery."

Victory - a vintage 1943 cardboard Egyptian-type dissected square puzzle

Mystic Wedge by the Crestline Manufacturing Co.of Santa Ana CA. 20 equal right triangles (10 black, 10 red) make a square. Derive this one by first cutting each of the five squares in half into equal rectangles, then dividing each rectangle along a main diagonal.

Dicksinson's Seed Ten Card Puzzle - 1910
Another variant of the Egyptian puzzle, similar to the Devoe 12-piece, but some right triangles have been fused to form two isoceles triangles. (I don't have this.)

Other Geometric Dissections


Super Star - Melissa & Doug
This is a dissection of a five-pointed star, in a tray.

Broken Heart
Form a heart from the 9 pieces.

Doctor's Puzzle Board

IQ Circle (PeToy Hong Kong)

Mind Bender Circle

Squaring the Circle - Dollar Tree

Perfect Squares

Profound Round
One of Mattel's Brain Drain series.

Fit the six pieces into the case to form a rectangle such that it contains only 3 straight seams. From puzzle-factory.com.

Form a six-pointed star using the six pieces. Also from puzzle-factory.com.

This set of "What's Your Score" puzzles from Shackman includes a dissected cross, square, and form a star.

Watney's Red Barrel puzzle
Build a red barrel from the pieces. A nice symmetric dissection.

"Jeu de la Croix" is a vintage French boxed version of a dissected cross on a pedestal.

"La Cocotte" is a vintage French boxed puzzle - form a bird shape from eight isoceles right triangles.

Bibendum six-piece rectangle

"Jeu de l-Octogone" is a vintage French boxed dissection of an octogon into 12 pieces. (I don't have this.)

The "Red Cross" or "Mysterious Cross" puzzle has been issued by several manufacturers of different nationalities and is known by various names. The eight red pieces form a Greek cross. The eight white triangles fill in the corners of the square.

IQ Mega-Form Circle

The Land Puzzle
You are given a 2x2 square, with one corner unit square missing, leaving three unit squares. Cut the shape into four identical pieces.

Stacked Triangles - George Miller

Stacked Squares - George Miller

Flying Saucer, designed and exchanged at IPP32 by Jeremiah Farrell, made by Chris and Walt Hoppe

Nightmares, designed by Jeremiah Farrell, made by Walt Hoppe
Would have been exchanged at IPP32 by Thomas Rogers (deceased)

PEKE - designed by Kohfuh Satoh
Form a Greek cross.
made by Saul Bobroff at Here to There Puzzles of Beverly MA.
a gift from Saul - Thanks!


Spear's Shape Puzzles

Squaring the Circle Jigsaw Puzzle - Copyright 1967 American Publishing Corp. Waltham, MA
Not really a jigsaw, since the pieces do not interlock, and each pieces' edges do not uniquely identify its neighbors.
Also not really "squaring the circle" - it simply comes with four identical curved pieces to be applied at the edges of a square, trivially making it into a circle.
The challenges lies in forming a 9x9 square using the 11 polyomino pieces whose total area equals 81 units.
Sometimes the objective is just to make a symmetric shape from the pieces (often a shape with just one axis of symmetry)...

Symmetrick, designed by Vesa Timonen, made and exchanged at IPP32 by Tomas Linden

Balance of Power, designed, made, and exchanged at IPP32 by Rod Bogart

Symmetric Shape, designed, made, and exchanged at IPP32 by Emrehan Halici

Trapezoid Symmetry - designed by Yasuhiro Hashimoto
Form 3 symmetric figures using only the two smaller pieces.
Then form 5 symmetric figures using all 3 pieces.

Dissected Letters

The dissected T has certainly been the most popular, but other letters have been dissected, too.


Missing T - Thinkfun
This is a version of the classic 'T' dissection, by Thinkfun.

Another classic T.

Pa's T from Drueke.

This cardboard version of the classic T dissection puzzle is a promotional item for a magician.

Chase & Sanborn Coffee-Tea
Showing both sides of each of the four pieces, which form the usual T.

An H dissection puzzle was included in the vintage "Deluxe Puzzle Chest No. 3006" from F.A.O. Schwartz.

A political promotion - form the letters F and D.

The "Famous F" puzzle
Note the trapezoidal piece - these pieces ar pretty much the same as in the "FD Puzzle."

Cracker Jack F
(I don't have this.)
Similar to the "Famous F."

Fletcher's F - an advertising promotion. (I don't have this.) Different than the "Famous F."

Furnas - The New F
Form an F from the six pieces.

Magic Z
  
Dad's Boy K
(I don't have this.)
I've drawn the four pieces.

An H Puzzle designed by Tomas Linden and made from Marblewood by Eric Fuller.
 
LinkBelt M Puzzle
 
I was asked for help in solving The New B Puzzle from Gold Star Coffee. Seven pieces form a letter B. I don't have this, but I did figure out a solution.

Form the word THINK from the 21 pieces.
The pieces of each letter are easy to discern since the letter to which each piece belongs is embossed on its face.
The T is the classic T dissection. The H is also familiar. The I is trivial. N and K gave some challenge.
I also found a copy in its original package.

New*T, designed, made, and exchanged at IPP32 by Nick Baxter

The vintage Celestial Cross puzzle issued by McLaughlin Bros. of NY.
   
A vintage five-piece cardboard Number Nine Puzzle, issued by the National Carbon Division of Union Carbide and Carbon Corporation, advertising Eveready Batteries. I have obscured the borders of the individual pieces in the photo of the solution.
 

The vintage Red Cross Puzzle is a twelve-piece dissection of a cross (or the letter t).
The small instructions slip identifies the source as The Gamo-Jig Company of 33 North Charter Street in Madison, Wisconsin.
The pieces are heavy cardboard - the backs are plain, not red, so the pieces must not be flipped over.
I have solved this one.


A dissection of a Rupee sign, designed and made by Scott Elliott

Transformations

Greg N. Frederickson is an expert on dissections which transform one shape to another, and discusses them at length in his 1997 book Dissections: Plane & Fancy.


Woodn't Tri - Reiss
Form a square from the 4 pieces. Then form a triangle.
This is a well-known dissection, originally called the "Haberdasher's Problem" and created in 1907 by Henry Dudeney. Discussed by Frederickson pp136-8.

Devil Puzzle
This set of pieces can also be put together to form a rectangle. It was offered by Bits and Pieces. It was also offered as part of a series by Nob Yoshigahara. This is the same set of pieces as in the Anchor Kobold puzzle.

Dudeney's Zoo from Archimedes' Lab
The triangle, pentagon, hexagon, and octagon are each dissected such that the pieces of each can form the square. 170mm x 120mm.

The Adams' Square and Cross.
Form a square or Greek cross from the four pieces.
(I don't have this.)

Form a square or a Greek cross from the six pieces. An advertising premium from Molson - the pieces are nice 1/8" plastic. Note the similarity to the Adams Square or Cross - two pieces have simply been divided.

"A Double Puzzle."
A vintage advertising puzzle from Dickinson's. It is the same puzzle as the Molson Square or Cross puzzle.

Mond oder Kreuz (Moon or Cross, aka Crescent or Cross)
Make both a crescent moon, then a Greek cross from the pieces.
From Wil Strijbos at IPP31 in Berlin. Thanks, Wil!
A nice wooden version of Sam Loyd's Cross and Crescent dissection/transformation between the crescent and a Greek cross (plus sign). Notable because of the curved edges accommodated. Notice the flattening of the tips of the crescent. The nice 7-piece dissection shown was actually found by Harry Lindgren. It avoids thin slivers and differs from Loyd's solution. Discussed by Frederickson on pp167-9.

Cut Out Puzzle
You are given a 2x3 rectangle, with one corner unit square missing, leaving five unit squares. Cut the shape into three pieces, which can be re-arranged to form a square.

Spade and Heart
by Mineyuki Uyematsu
Make a Spade or a Heart from the four pieces.
Purchased at IPP28 in Prague.

A vintage Cracker Jack premium - the "Chicken and the Egg Puzzle"

T+3 - designed by Hiroshi Yamamoto
The 3 pieces can be arranged to form four different pentominoes, including a T.
A really nice dissection!
This won a Jury Honorable Mention at the 2011 IPP Nob Yoshigahara Puzzle Design Competition.

Checkerboard Dissections

Over the years, there have been many variants on the theme of a dissected checker- or chess- board. Jacques Haubrich has published a compendium of checkerboard puzzles in two volumes. The first volume, "A Century of Checkerboard Puzzles," describes all known checkerboard puzzles - over 440 of 190 different types - published between 1880 and 1980. The second volume, "Additional Checkerboard Puzzle Designs," covers checkerboard puzzles published in the last 25 years.
Jacques characterizes the puzzles using a code of the following format and meaning:

N[2].D.S-L

  • N is the number of pieces
  • 2 is present if the pieces are 2-sided
  • D is the number of different piece types employed
  • S is the number of squares in the smallest piece
  • L is the number of squares in the largest piece


Slocum and Botermans, in their 1986 book "Puzzles Old & New" suggest that
the first checkerboard puzzle was this Sectional Checkerboard of 15 pieces,
patented in 1880 by Henry Luers (231963) and produced by Selchow and Righter.

This "Krazee Checkerboard Puzzle" was made by The Plas-trix Co. Inc. Jamaica NY. There is no date on it, though in Haubrich's "Century" the date listed is 1957. This variation has code 12.11.3-7. My Dad had a puzzle like this, but it's gone - and I don't remember which variant it was.

"The Famous Checkerboard Puzzle" - "Only 12 pieces but - Oh My!"

This is the Bug House puzzle. Jacques gives a date of 1912. This has metal, rather than cardboard, pieces. It was offered in a rectangular box, and in a square box. It has code 14.14.3-5.

This one is the "Unique Original Checker Board Puzzle" from the Unique Novelty Company, and not only is it "Improved" but it is also the "Most Difficult Puzzle Known." It has code 14.14.3-5 and is the same set of pieces as the Bug House. No date is given in Jacques' book, but Slocum and Botermans bracket this in 1930-1940.

This one, "manufactured by J. F. Friedel Co., Syracuse, N.Y." calls itself "The Original Checkerboard Puzzle." I have no idea if the claim is true. There are 15 pieces and the price on the box says 10 cents. Jacques gives no date. Code is 15.14.3-6.

The Famous and Baffling Checker Board Puzzle has fourteen pieces and originally cost fifteen cents. 1927, code is 14.14.3-5. Inside the cover, the Vasen Mfg. Company of Davenport, Iowa, ran a contest offering $500 in Gold for the greatest number of correct solutions. Unfortunately, the contest expired July 15, 1928.

Checkle Heckle is a checkerboard dissection in the Mattel Brain Drain series from 1969. It consists of a tromino, 4 tetrominoes, and 9 pentominoes. The pieces cannot be flipped, so some mirror images are included. This is the same set of pieces as the Famous and Baffling Checker Board Puzzle. Code is 14.14.3-5.

Angle Mania has 15 pieces, but only 14 are needed to complete the puzzle. Four different pieces can be left unused. From 1984. Code: 15.15.2-6.

This is a recent wooden variation called just the "Chess Box." It includes a set of 12 checkered pentominoes plus a 2x2 checkered square piece.

Golf Tease - Great American Puzzle Factory 1996. Assemble 14 pieces into a 9x9 checkered square.

An advertising puzzle of 14 polyominoes from AMF.

The older version of the TSL Draughtboard Puzzle.

The newer version of the TSL Draughtboard Puzzle.

But - Oh My!

An advertising puzzle for the Burlington Railroad.

All Square

Uneasy Checkers

Tenyo Checker puzzle

Tenyo Checker puzzle (back)

A vintage 12-piece Banzee Island Checkerboard Puzzle. An advertising promo for McCracken Realty Inc of Phoenix AZ.

Sectional Checkerboard Puzzle - a vintage advertising promotion from the Phenyo-Caffein Co. of Worcseter MA. "Patented Sept. 7, 1880."
This is the 15-piece Luers design.

Phenyo-Caffein Checkerboard cont'd

Phenyo-Caffein Checkerboard cont'd

Andy Snowie's CalmPlex MindBlock is part checkerboard dissection. I made one from LiveCube.

Slocum and Botermans in "Puzzles Old and New" on page 14, and also Ishino's site, describe another variant (I don't have) from 1908 called Broken Chessboard by Henry Ernest Dudeney. Since it is composed of 12 pentominoes and a 2x2 square, as is the Chessbox above, they might be the same.

Here is the 14 piece "Cut-Up Checkerboard" from Edwin Wyatt's 1946 book Wonders in Wood (I have highlighted the piece borders):

Dissected Dice

There are several cubic puzzles in the form of a dissected die. In Hoffmann's Puzzles Old & New, The Spots Puzzle is number XVII in chapter III. The puzzle consists of nine 1x1x3 bars, each decorated with some pattern of spots (pips on the die). The task is to assemble a 3x3x3 replica of a die, having the correct arrangement of pips on all six sides. The modern puzzles below are all based on the same principle.
You might want some clues - a valid die has the following attributes:
  • there are 21 pips in total
  • opposite sides sum to seven
  • Here are some additional characteristics that might vary from die to die...

  • the two rows of three spots each on the six face are aligned with the corners of the four
  • with the six up, the two slants from upper left to lower right
  • with the six up, the three slants from upper right to lower left

    +---+
    |  *|
    | * |
    |*  |
+---+---+---+
|  *|* *|* *|
|   |* *| * |
|*  |* *|* *|
+---+---+---+
    |* *|
    |   |
    |* *|
    +---+
    |   |
    | * |
    |   |
    +---+


Intelligence Puzzler

Cracked Dice - Lakeside 1969
There are three dice - one whole (serves as a prototype) and the other two dissected into three 1x3x3 pieces each.

Make a Dice Puzzle
Can you solve in 8 minutes?
Copyright 1957 St. Pierre & Patterson Mfg. Co.

The Broken Die - Gantt's Wood Things

made in China

Twice Dice - Pentangle
(small version)

Twice Dice - Pentangle
(large version)

Woodn't Die - Mag-Nif

I found another dissected die - it seems fairly old - it comes in a purple box and has nine red pieces with white pips.
I am unsure of the provenance - there are no markings on the box or pieces.
The box is 1.75"^3 (45mm^3).
There should be at least 21 pips, but there are only 20 - so evidently one is missing.
The pieces are not the same as the Wolff Spots Puzzle described in Hoffmann.


A vintage Shackman "Dice" assembly puzzle, with instructions
 
"Vegas Baby" cube from SiamMandalay
(A gift from my brother.)

Weave Puzzles, or The Crossed Sticks Family

I bought an "8-block Collusion" puzzle from Rocky Chiaro. Rocky refers to the Collusion and its relatives as "pin puzzles." I solved Rocky's Collusion and realized it was similar in principle to several other puzzles in my collection such as Jean Claude Constantin's "The Fence" that don't necessarily employ pins.

I call this group of puzzles the "Crossed Sticks Family." A set of rods/sticks are crossed in two layers, with the points where each rod crosses (mates with) another constrained by a feature present at that location on the rods, and the compatability of the respective features. The crossings define a grid. Identical overall physical dimensions make the rods interchangeable (except for their features), and features are positioned at crossing points. The notching positions are well-defined along the rod, and the number of potential notch positions is related to how many rods cross.

When rods can be inserted only one direction into a frame, I call them "asymmetric."


The progenitor of this family seems to be this puzzle called Sputnik, made in the 1950s in Japan.
There was also a version from 1958 with six sticks called the "Mysto-Peg Puzzle."
Sputnik is described on page 59 of Jerry Slocum's and Jack Botermans' 1987 book "Puzzles Old and New."
Rocky says it was his inspiration for his pin puzzles.

Notched rods can be assigned unique identifiers simply by giving them a binary code - start on one end and compose the code with a zero for no notch and a one for a notch. For three kinds of features, e.g. holes, flats, and pegs, count in trinary, etc. When determining the ID for a piece, my convention is to orient it so that the "endmost" notch is rightmost, and number with the LSB on the right.

The features are drawn from a specific set:

I wrote a program to analyze this class of puzzles. Click here to run my crossed-sticks puzzle solver program. It is written in javascript and runs client-side on your computer. (So IE 6 may complain about security - you might have to allow blocked content.)

First choose the "degree" of the puzzle - how many rows (or columns) to model (1-5), then choose the apropos number and type of pieces from the list. The program is limited to degree 5, and only two rod features. I don't support duplicate pieces, but several pieces are end-for-end symmetric so there is a way to "cheat." Scroll down the page and click "Run." Each time a solution is found, an alert box will pop up listing the configuration. You can hit OK to continue and find another solution, or Cancel to quit and inspect the current solution in the bottom-most graphical area labeled "Inspect."

Symmetry considerations: the program currently does not recognize all symmetries, so it will produce some redundant solutions. However, for a regular grid there are a limited number of distinct locations where the first rod can be placed. The locations can be divided up into a small number of distinct classes, and placing the first rod at any location within a class is equivalent to placing it at any other location in the same class. This allows one to choose an arbitrary single location within a class for the first rod and omit analysis of all other arrangements where the first rod would be in any other location in the same class. I call the classes the "equivalence classes" for the grid and identify them using A, B, and C. Furthermore, for a given set of rods all of which must be used, any rod may be chosen arbitrarily as the first rod to be placed.

You can also use BurrTools to solve this type of puzzle. I programmed in all 16 4-position asymmetric pieces for a 4x4 arrangement, along with a frame, and found 408 solutions using various sets of 8, disallowing duplicates and internal voids.

Two-Axis Arrangements, Two Features, 3x3 and 4x4, With a Frame

8-block Collusion - by Rocky Chiaro
An elegant Victorian-looking puzzle machined by Rocky from brass.
Asymmetric Pieces: 0, 1, 2, 6, 7, 12, 14, 15.
Five solutions (one shown).
 
Rocky sent me this 1-block version of Collusion he calls the "AB-L" puzzle,
after the person who inspired it.
Same pieces as 8-block, but symmetric.
During IPP26, Bits and Pieces held a reception and distributed a souvenir version.
 
Haba Crux
Asymmetric Pieces: 0, 1, 2, 6, 12, 13, 14, 15
One solution.

A nice hefty lucite 4x4 weave puzzle - I found it in Montreal.
Pieces: 0, 1, 2, 6, 12, 13, 14, 15. (Same as Haba Crux.)

This is a cheap monkeypod wood version called the "Snag Box," also known as the "Computer Chip."
Pieces: 0, 1, 2, 3, 6, 7, 14, 15.
Two solutions.

Here is another 4x4, called "Weaver's Dilemma." I don't own this puzzle, but I wanted to show it because it uses duplicates of several pieces.
Pieces: 0, 2, 2, 3, 6, 6, 15, 15

A wooden 3x3 weave puzzle from "The Akron Los Angeles CA 90038."
Made in Japan.
Pieces: 0, 1, 1, 5, 5, 7.
 
The Fence
Jean Claude Constantin
Pieces: 1, 2, 3, 5, 6, 7, 9, 11.
Two-Axis Arrangements, Three Features, 3x3, 4x4, 5x5, No Frame

Stabpuzzle, and Acht (8) Stabpuzzle (Logika)
In the original 3x3 version, each stick has one face with 3 square locations along its length - at each location, the height of the stick can be level 0, 1, or 2. Mate the sticks in 2 crossed layers of 3. Logika now also offers a 4x4 version.

I liked Stabpuzzle so much,I made a complete set of all possible pieces from Lego. There are 18 excluding end-for-end symmetries. 3 are flat - one at each level. 6 are end-for-end self-symmetric. The remaining 9 are asymmetric. Stabpuzzle uses 3 symmetric { 010, 121, 202 } and 3 asymmetric { 011, 102, 112 } pieces.
For Rob's Stick Puzzle, use the six asymmetric pieces not used in Stabpuzzle.

Timbers by Mad Cow
Pieces: 010, 120, 200, 201, 202, 220.

The Log Pile puzzle uses 10 sticks in 2 crossed layers of 5. There are 13 pegs and 13 holes, so no hole/flat or hole/hole matings will be possible.
Pieces: 00110, 01012, 02011, 02101, 02121, 10121, 11012, 11202, 11212, 12112.
Three-Axis Arrangements

The "Clive Cube" is a representative of this group, extending it from the 2 axes employed above, to 3 orthogonal axes.

The "IQ Puzzle" or "Ten Pins" puzzle is another 3D example.

I got this version from Torito. They call it "Sapience Sticks."

The Nine of Swords

This is the plastic Reiss version of the Nine of Swords.

Arjeu Achille CT5152
Purchased at GPP.
Special Arrangements

Haba Verticus - designed by Heinz Meister
Ten sticks, with holes or flats at 5 positions.
Arrange them in two crossed layers of 5 each,
such that holes in both layers all align.

Rick Eason's Keyhole Puzzle calls for 6 sticks to be arranged in a 3x3 sandwich, but with the additional complexity of sequential assembly. The pegs are screws and the holes are "keyholes" into which the screwheads must slide in the proper direction.

Rick has taken the keyhole concept into a new dimension
with his Keyhole Cube.

Prismazul Octuple - designed by Ingo Uhl, made by Logika Spiele
Exchanged at IPP31 in Berlin by Tanya Thompson, purchased from Tanya.
Build a triangular prism with the eight pieces - 4 equilateral triangles each with a footprint of 4 unit triangles, and 4 rhombuses each with a footprint of 4 triangles.
Each piece can have unit triangles at three different levels: L, M, and H. There are 11 unit triangles at L, 10 at M, and 11 at H.
The rhombuses preclude the target shape being a prism like a Toblerone bar, with a cross-section of an edge-2 triangle.
A triangular prism with a cross-section of an edge-3 triangle is also precluded since the available 32 unit triangles could not be distributed evenly.
(Each layer would have a footprint of 9 triangles - 3 layers uses 27, which is too few, and 4 layers uses 36, which is too many.)
That leaves a 2-layer triangular prism with a cross-section of an edge-4 triangle, giving a layer footprint of 16 unit triangles.
This implies that the pieces be arranged in two layers each containing 2 triangles and 2 rhombuses.
(The four rhombuses alone cannot tile a side-4 triangle, neither can one rhombus with 3 triangular pieces.)
If the pieces in the layers are oriented so that their multi-level faces face the opposite layer,
then the 10 M units will mate 5x5, and the 11 L's will mate with the 11 H's.
The indication that there should be 5 M units in each layer seems like a good clue!

Alexandre Muñiz contacted me regarding "crossed stick" puzzles. He has designed several, and very kindly sent me examples of his 10-piece Decagram, and his 8-piece 4-Pointed Star. You can read more at his website PuzzleZapper.com. Thanks, Ali!


You can specify what I call a "compatability matrix" for each puzzle, showing whether a given mating is permitted (Y), is prohibited (x), or is physically possible but might be excluded by a rule (?), or by the relative availability or distribution of the features of different types. Only the upper triangle of the matrix is needed.

Potential new puzzles suggest themselves - extend the set of features to 4 or more, and/or use a different compatability matrix. Use features other than levels, or pegs/flats/holes - how about magnets? A magnet embedded in a rod offers N or S, and the absence of a magnet at a position allows a third feature. I include the compatability matrix for an imaginary Rob's magnetic puzzle below. No magnet = 0, N = 1, S = 2.

Collusion
  0 1
0 x Y
1   ?
The Fence
  0 1
0 x Y
1   x
Timbers, Log Pile
  0 1 2
0 ? ? Y
1   Y x
2     x
Stabpuzzle
  0 1 2
0 x x Y
1   Y x
2     x
Rob's Magnetic Puzzle
  0 1 2
0 Y Y Y
1   x Y
2     x


Prairie-Dog Town, Alien Hive, and Tee Time each include 6 pieces (they're really the same puzzle) that must be arranged in a 3x3 "sandwich" between two 3x3-holed plates. Prarie-Dog Town is fun to analyze and solve.

These puzzles are slightly different from the previous examples, in that a piece can have "pegs" (or bumps) on both sides.

The puzzle uses six planks of width 1 and length 3 units, and includes two 3x3 plates with all holes. At each of the 3 positions on a side of a plank, there may be a hole (which goes through the plank and therefore also appears on the other side in the same position), a flat, or a peg (which fits into a hole). When the pieces are mated, a peg must mate with a hole from one side, and this blocks BOTH sides of the hole. Two holes, two flats, or one of each may also mate. So a length-3 plank has 6 positions at which a hole, flat, or peg could exist. For any plank, there may be a maximum of 3 holes or 6 pegs.

Below is my enumeration of all possible pieces for this style of puzzle. The Prairie-Dog Town puzzle utilizes 6 pieces, outlined in red in the chart (the piece with 2 bumps and 1 hole, with the hole on an end, is used twice). I have shaded green the cells containing only pieces having at least one pair of opposing pegs.

Sequential Assembly

I have added this category because there are several puzzles that one can argue belong in the assembly section, since they are neither true interlocking puzzles, nor are they true sequential motion puzzles where long operators are needed, yet they do require some back-and-forth or rotational movement of the pieces, and the order of placement of the pieces can matter.


Crossroad - designed by Goh Pit Khiam and made by Walter Hoppe. Purchased from Walter at IPP28 in Prague.

Pack the Square, designed by Goh Pit Khiam, made by Joe Sarabande, exchanged at IPP32 by Larry Seidman

The ODD Puzzle - designed by Hirokazu Iwasawa (Iwahiro).
Three pieces (two identical) to pack into the box. Winner of the "Puzzle of the Year" Award in the IPP28 Design Competition.

Eight Pack - designed by Tom Jolly and issued by Philos
Pack eight tetracubes (four tower-left and four tower-right) into a 4x4x4 cage.
Purchased from a puzzle store in Prague.

Here is a set of pieces in a cage. I received this puzzle in a trade with P. F. Ramos - he designed it and Interlocking Puzzles (now defunct) made it. It is called Twin Pentominoes Into a Light Box. There are two instances of each of the non-planar pentomino pieces.

Pentominoes in Cage
from Brian Menold at Wood Wonders

Closterman cube
Six pieces fit sequentially into a cubic cage. Nicely handmade in Yellowheart wood.

Mine's Cube in Cage
from Brian Menold at Wood Wonders

Framework II - designed by Markus Götz
Made by Eric Fuller, from Walnut, Sapele, and various other exotic woods.
A redesign by Markus of Markus' original, with analysis by Tom Jolly. This version has a single solution. The pieces fit into the tray which is open on both sides. Offset tabs attached to the piece blocks grip both the frame and other pieces, making this somewhat interlocking and requiring sequential [dis]assembly.

Brunnenspiel, by Markus Goetz

Circelei - Hendrik Haak IPP26
Fit three hinged 3-layer polyominoes into three stacked trays.

This is the Lolly Box, designed by Alfons Eyckmans and made by Eric Fuller - a walnut box, with Bubinga, Pau Ferro, Purpleheart, and Paduak pieces.

On 6/29/13 I had fun attending a puzzle party hosted by Saul and Paulette Bobroff at their home. Several notable puzzlers were there and we all got to play with much of Saul's extensive puzzle collection. Saul kindly gave me one of his IPP31 exchange puzzles - design number 138A by Stewart Coffin - the Piggy Box. Thanks, Saul and Paulette - I had a great time!

Core - designed by Yavuz Demirhan, made by Eric Fuller
from Sapele and Cocobolo.
Fit five pieces into a 3x3x3 cavity in the frame, working through a 2x2x2 opening.

Boussole - designed by Yavuz Demirhan, made by Brian Menold
from Poplar and Cocobolo
Fit four pieces into the frame.

Odds and Ends

Many assembly puzzles use pieces themselves constructed from regular units - cubes, spheres, or tetrahedrons. An assembly puzzle can also have irregular dissimilar pieces. This section covers a wide variety of unusual assembly puzzle designs.

Each Paracelsus Puzzle is one of a series of unique castings. I have three - a Disk, a Oil Drop (or Waterfall), and Birds. The material is silicon bronze. These were made by Steve Johnson of Port Townsend, WA. Purplepomegranate.com used to list him as one of their artists as late as Sept 2004, but no longer. Maybe he’s out of the business?

I found the following photo of a catalogue page showing seven designs:

I received a request for help in assembling the Waterfall puzzle. Even though every Paracelsus Puzzle instance is unique, the following step-by-step assembly images of my copy might help:


Penthouse from Pentangle

Screwball (an oldie)
U.S. Patent 3813099 - Scott 1974

The "Moron Puzzle"
To quote from the label:
Morons - Take 1 Min.
Idiots - Take 2 Min.
Goofs - Take 3 Min.
Numbskulls - Take 4 Min.

Bamboozle - B&P

5th Chair - Thinkfun (Gift from Brett)

The Chaotic Cube

2 Scheibenpuzzle (Logika)

Heart - Logika

4-piece puzzle - Logika

Think Tac Toe - Pressman

Pegged - B&P


Olistripe
The pieces interlock somewhat, but not enough.
See U.S. Design Patent D500347 awarded to Daniel R. Oakley in Dec. 2004.

Think|a|ma|jig
Copyright 1974 by Leonard J. Gordon (Gordon Bros.)

Jumpin' Frog Jumble
The pieces do "interlace" but they don't really interlock in a solid 3D structure.

The Woody Cube (Nankai) - B&P

The Intragon from Naef
Designed by Jost Hanny in 1989.
Twelve pieces assemble inside a frame. See the Intragon pieces here.

Six Key Mine (B & P)
An R.D. Rose design. First Prize, 2003 IPP Puzzle Design Competition. The pegs have tongues that can interfere inside the sphere. Insert all 6 without interference.

Just Fit - William Strijbos
16 pieces plus tray. Create a two-layer 5x5 checkerboard in the tray.
1990 Hikimi Wooden Puzzle Competition winner.

Diamond Mind - Constantin

Diamond Soul

Hippo Haven (Thinkfun)
Each Hippo has two pegs. The pegs and holes in the base are of 4 different depths. Find a way to fit the Hippos in the base so all pegs are completely inserted.

Short Circuit
Purchased the Constantin version at GPP.
Similar to Hippo Haven.

Hooked Cube
Philos (Goetz)

Juha Six J's Cube set - IPP19
Together, the 24 pieces from the four cubes can make over 200 assemblies.

Tower of Babble by Leonard J. Gordon Item No. 134

The Infernal Triangle was issued by Gordon Bros. and is marked "Item No. 135 1974 Leonard J. Gordon." The seven pieces are similar to those of the Tower of Babble, but here you must arrange them to form a two-layer triangular grid with 5 cylinders along a side.

Surface

Harry Potter Mirror
(see U.S. Patent 6976678 - Setteducati 2005)

Punch Cards
Tom Lensch

A set of McDonalds promotional puzzles

Link Puzzle
make a cube from the loop of chain links

Rising Mountain

This is a sculpture made of South Australian Red Gum wood by Robin Turner. I believe it is one of his "Ayers Rock" series.

Impuzzleble

A set of vintage puzzles from Plas-Trix of Brooklyn NY, includes: Krazee World, a pair of Batee Baseball, a pair of Krazee Links, a checkerboard dissection, a dissected scene

Nuts and Bolts - Learningsmith

Tool Trouble
1996 Great American Puzzle Factory, Norwalk CT.
Assemble the 17 irregular pieces into a 7" x 9" (4x4 piece) rectangle. Six of the pieces have diagonal edges. Each piece depicts some tools, but they have nothing to do with the solution.

Prismentwist - Logika

Tuned In
Milton Bradley 1973
Using all 14 gears, assemble a gear-train linking the knob with the male and female symbols.

Chess Cubes

Daily Mail Crown Puzzle
There have been several puzzles produced based on the theme of two sets of copies of distinctly-shaped pieces, where one set can be used to completely cover the other set (i.e. the sets cover the same area). Some of these are very challenging!

Cover It Up
Designed by Robert Reid; this was Saul Bobroff's exchange puzzle at IPP26, where it won an Honorable Mention in the Design Competition. Cover the dark pieces completely with the light pieces, no overlapping the darks. The total area of dark and light each equals 4x7=28 units. It should be possible...

Boston Cover Up - designed by Robert Wainwright

Top This! - Thinkfun
This Thinkfun puzzle offers a set of challenges similar to Cover It Up and Boston Cover Up, but simpler.

A Cover-Up variant by Krasnoukhov
Purchased at IPP 29 in SF

Erich Friedman's Cover Up design - three challenges.
From Creative Crafthouse

MetallWürfel - Constantin

Times Square - B & P

Ziggurat - Creative Crafthouse has it.

Dizzy Tower - Dizzy Art 1996

Naef's Discon puzzle, designed by Jost Hanny.
Also, Discon Fever - a copy of Discon from B & P
Peter Kaldeway's site shows a solution.

I received this nicely made copy of the Discon puzzle, from craftsman Steve Kelsey. Thanks, Steve!

Barricade - B & P

This is Mental Block Puzzle #5 Vortex, by R. D. Rose. It is crafted from aluminum and comprises five rings with various pegs and holes around their perimeters, which must be assembled into a cylinder.

MT5T (Make the Five Tetrominoes) - Mission 1
designed by MINE (Mineyuki Uyematsu)
Arrange the four large pieces so that a subset of their gaps exactly enclose the five tetromino pieces.
A similar version won a Jury First Prize at the 2011 Nob Yoshigahara Puzzle Design Competition

Idea Cube - by Idea Ocean

A paper version of Deep Sea Tango - obtained from George Hart at the 2007 NYPP.

The 3Q Cube designed by Takeyuki Endo. Fit the three two-cube pieces into the cage. 2 solutions.

Milton Bradley made a couple of "Stickler" puzzles. Insert pins into a stack of disks which have holes at various points. The disks must be aligned so that all pins can be inserted.

Schalenwurfel - Logika

Keiichiro Ishino modified Takeyuki Endo's 3Q Cube so that it has only one solution. A gift from Bernhard Schwietzer, at NYPP 2008. Thanks, Bernhard!

A selection of "Dicebox Mindbender" puzzles
by Mi-Toys -
Half-Cubes, Rod by Rod, and Stacked Sticks, purchased at Eureka, and Cube Mates, from Brett.
Imported from China by CHH Games.

Three diminutive but colorful plastic puzzles from Germany - build a cube from six panels, build a cube from nine concave tricubes, and build a step pyramid.

9-Post Packing Puzzle
De Vreugd
B & P

IQ Cube - Brainbenders
Eight cubes with tabs and slots. Make a 2x2x2.

This is a relatively inexpensive mass-produced copy of Wayne Daniel's famous All Five assembly. Purchased from Mr. Puzzle Australia.
Here is a series of assembly puzzles by Andy Snowie:


From left to right, they are: Orbsticle, ConeFusion, CyliPlex, EllipToy, and Pocket CalmPlex.


Jamaika - by Markus Goetz

Tirol Chocolate
Purchased at IPP28 in Prague, from Wil Strijbos.

Octix - Trigam

Pairs of Prisms
Ergatoudis IPP13 exchange

Trevor Wood's Prism Cube - unknown craftsman
Made from highly figured canarywood.

3 Pyramid Cube by Philos

The Jeu du Cube and L'Enervant puzzles are vintage French non-cartesian cube dissections. (I believe Le Tracassier is also the same set of six pieces.)

Obsivac Cube 1

Obsivac Cube 3

Naef Kniff
by Manfred Zipfel and Cordula von Tettau
(See Ishino's Kniff page.)
Purchased at IPP28 in Prague.
The L-Ements series by Rick Eason. In each case, build a cube from the elements.

Seven L-Ements IPP25

Eight L-Ements

Nine L-Ements IPP23
See Ishino's page on the Nine L-ements puzzle.

eL Perch
Rick's exchange puzzle for IPP 29 in 2009 in SF
Build a cube from the seven pieces such that it stays together on the tripod "perch."

8 L-ements - designed by Rick Eason
from Creative Crafthouse

The Triangle Cube
aka Pantene

The 3456 Pythagoras Puzzle from Pentangle challenges you to use the nine pieces to form a set of three cubes 3x3x3, 4x4x4, and 5x5x5, then add them together and form one 6x6x6.

Hexahedroom
This very nice puzzle was made by Eric Fuller, from Ebony and Jatoba woods. Form a cube within the box by fitting the pieces in via the available holes. A cool solution. Based on an IPP25 exchange from Hirokazu Iwasawa.

Olymp by JCC

The Double Octagon Box from Bits & Pieces
Same idea as the cereal box puzzles from Synergistics.

peg square
(not sure of name or manufacturer; it's not the Naef design)

The Sticky Cube
Designed by Bernhard Schweitzer
A gift from Bernhard at IPP 29 in SF - thanks!

This six-piece puzzle is a 3-D printed adaptation by George Bell, of Stewart Coffin's Peanut design (see the original in wood at PuzzleWorld, and at Scott T. Peterson's site). I ordered the 3 cm. version from George's Shapeways store, in Alumide material. (Photo by John Devost.)

Overall, According to George Bell, the six pieces (or subsets of them) can form only a limited number of symmetric shapes (but he doesn't know how many of each could actually be assembled, since in some the pieces interfere with each other and won't slide together):

(1) 3-ball: 2-1 flat triangle;
(4) 4-ball: 2x2 flat square; diamond; "canoe" (almost corners of a tetrahedron)
(5) 5-ball: 4-1 square pyramid; "bridge" (2-3 flat trapezoid); 2x2 square w/ one outlier;
(19) 6-ball: 6-ball flat ring; 3-2-1 flat triangle; "arch" (2x3 chevron - NOTE: the 2x3 rectangle is impossible); "boat" (flat 2x2 square w/ two opposite outliers); "cannon" (4-1 square pyramid w/ one outlier); octahedron (2x 2-1 flat triangles, packed vertically); "squashed octahedron" (2x 2-1 flat triangles in two offset layers);

There are many more asymmetric shapes.


Only 2 Sticks designed by Kofuh Satoh and made by Saul Bobroff
Purchased at NYPP Feb. 2010

Holzwurm (Product No. 6038), from Philos.
Designed by Dieter Matthes.
Form a 3x3x3 cube from 9 pieces having protrusions and hollows.
Purchased at The Games People Play.

Twist-L-Dan, in Oak, Wenge, and Karin woods,
designed by Takeyuki Endo.
Purchased from the Karakuri Club.

8Pd, in Oak, Angsana, and Karin woods,
designed by Takeyuki Endo.
Purchased from the Karakuri Club.

One Four All & All Four One
Made by and purchased from Mr. Puzzle Australia. Designed by Arcady Dyskin and Pantazis Houlis
Entered in the IPP30 2010 Nob Yoshigahara Puzzle Design Competition, where it placed in the top ten.
Arrange the four pieces (representing the three Musketeers plus d'Artagnan) in the frame so that they are self-supporting in the frame - you must be able to handle the frame without the pieces falling out.
The frame is made from Queensland Blackbean with Queensland Silver Ash joins.
The pieces are made from Queensland Silver Ash, Papua New Guinean Rosewood, Western Australian Jarrah & Queensland Blackbean.

Yubisaki Annai - Takeyuki Endo - IPP30
Fit the six 1x1x2 blocks into the cage. Five blocks each have a protrusion that will interfere with other blocks and the cage.
A set of three Star Wars themed Pizza Hut (South American) promotional puzzles from 1997 -
the Death Star (interlocking/assembly); Han Solo frozen in Carbonite (sliding piece); and R2D2 fixing C3PO (dexterity):


Promotional puzzle from IBM
What solid shape will fit through each hole,
completely filling the outline?

Tri-Bal Trifle, designed and exchanged at IPP32 by Rob Hegge, made by Formulor
Assemble the pieces so that the triangular armature balances.
 
Knobeltorte
A put-together puzzle in an egg. Layers of pieces with indents and knobs similar to the Prairie Dog Town puzzle.

Full Bloom, designed by Ferdinand Lammertink, made and exchanged at IPP32 by George Miller
Eight rings, each having two petals in various positions. Stack the rings so that no petals overlap.

Jerrymander, designed and exchanged at IPP32 by Bill Cutler, made by Laser Perfect

Washington DC Sightseeing, designed by Tania Gillen, made and exchanged at IPP32 by Marcel Gillen

DC Tease, designed, made, and exchanged at IPP32 by James Kerley

Tantalizing, designed, made, and exchanged at IPP32 by Yee-Dian Lee

Shameful Congressional Gridlock, designed and made by Vaclav Obsivac, exchanged at IPP32 by Patrick Major

Brain Blocks by Winning Moves
Eight differently-shaped blocks have various detents and tabs on their sub-faces that constrain how they can be abutted. Use the blocks to form target assemblies.

Bill and Betty Bricks - designed by Raf Peeters, issued by Smart Games
A set of wooden blocks and two figures. For each challenge, set up specific blocks as the base and stand one or both figures on top. Using specific additional blocks, build up a rectangular structure, always moving the figure up only one floor at a time. Decorated in a playful motif, but including both simple and advanced challenges!

Square in the Bag - designed by Hirokazu Iwasawa - a nice wooden square with a cloth bag.
Find a way to cover the square completely with the oblong bag.
This puzzle won the Puzzler's Award at the 2012 IPP Design Competition.

I ordered this and Cor-RECT-ly in the Bag from MINE in Japan, based on a post on Jeff Chiou's blog.


Cor-RECT-ly in the Bag - designed by Hirokazu Iwasawa

Puzzle Rings

Puzzle finger rings (Wikipedia article) made from several interlaced bands have been crafted by artisans from many cultures, and date back many years. The Puzzle Ring Store has a lot of info and a solution library.


This 4-band puzzle ring was included in the "De Luxe Puzzle Chest" No. 3006 from F.A.O. Schwartz. It's the Extraordinary Ring Puzzle No. 3522 by Shackman. Made in Japan.

This 6-band puzzle ring was designed by Bram Cohen. It's 3-D printed. I bought it from Bram at IPP 29 in SF.

Here is a 7-band puzzle ring I got in an auction lot.
 
Holistic Ring, designed and exchanged at IPP32 by Bram Cohen, made by Oskar van Deventer & Shapeways
 

Puzzles from Synergistics Research Corporation

An outfit called "Synergistics Research Corp." (New York, New York 10011), which evidently no longer exists, made several plastic assembly/packing puzzles years ago. I have not found an exhaustive list, but they include:

(Synergistics also made various jigsaw puzzles, covered in my jigsaw section.) I've obtained all of the above...


Here is an analysis I did of the Synergistics LifeSavers puzzle. I have found that all "flavors" use the same set of piece shapes. Each consists of 12 tori having combinations of pegs and holes. The tori stack together and fit into a cylindrical container approximately 55mm in diameter by 120mm high.

Each torus has a central hole immaterial to the solution. Each of the four cardinal positions (i.e north, south, east, and west) on its two faces may have one of the following features:

In total, there are 22 holes and 22 peg-lengths. For a puzzle of this type to have a solution, the total holes must equal or exceed the total peg-lengths.

This puzzle can be solved using PuzzleSolver 3D, if it is mapped to an analogue composed of unit cubes. My mapping is straightforward but imperfect as it will allow "illegal" solutions - fortunately the first solution produced is acceptable.

My mapping is as follows: rotate each torus depicted by 45 degrees clockwise. Use a 3x3 grid of cubes to model the torus and any holes - delete a corner cube corresponding to any hole. Leave the center cube filled in, to ensure the piece remains contiguous as required by PuzzleSolver. Add a cube extending outwards for a peg, or two stacked for a double-length peg, at the appropriate corner positions on either side. The target volume is 3x3x12.

The diagram shows the disks, and the mapping of each disk to cubes. Two pieces are duplicated.

Solution number 1 is: 5, 2, 12, 9, 8, 6, 3, 1, 10, 11, 7, 4.

And here is an image of the solution #1, clipped from Puzzlesolver 3D:

 

Puzzle Food

Synergistics isn't the only firm that made puzzles in the shape of food items. Here are some additional examples...


I had a Parker Brothers' "Phony Baloney" when I was a kid - it disappeared but I found one in auction.

This miniature version of Phony Baloney was a cereal-box prize.

Here's one I found called the Banana Split, by Lakeside.

Here is another Lakeside puzzle with a food theme - the Apple. Assemble the eight slices around the core so that the two "worms" can be inserted through the core to hold the puzzle together. The pieces have holes at 14 different heights, only two of which will line up with corresponding holes in the core. There are only two pairs of slices having core-aligned holes. Not difficult, but cute.

Prankfurter - Reiss

Burger Thing - Reiss

Here is a puzzle chocolate bar, the "Puzzle Bar" from Pentangle.

Another hamburger puzzle, made in China.

Here are Peter Piper's Fickle Pickles, a ten-piece packing puzzle.
Made in Hong Kong, copyright 1973 Steven Mfg. Co.
Discussed in Slocum and Botermans' The Book of Ingenious and Diabolical Puzzles on pp90-91.
(Click the image to see the solution, cheater.)