These are puzzles that rely on magnets either to stay together or as part of their mechanism. (I am excluding hiddenmechanism types using a magnet to hold a pin in place.)
Stickman Chopstick Box (#13, Second Edition) by Robert Yarger Purchased from Cubicdissection. Beautifully made from paduak and bloodwood, with maple inlays. Seven moves are required to release the two chopsticks, which are themselves beautifully inlaid. 
Magnetic Bumper Cars  Popular Playthings / Huntar Co. Inc. 2006 
Kayak Cove, from Popular Playthings. 

The DigiDisc is a series of tori with mathematical symbols along their peripheries. Arrange them so that all equations are true. I wrote a computer program to solve DigiDisc. 
The Magnetic Puzzle Ball from Executive Minds contains a central sphere  each piece has a stem ending in a magnet that attaches to the sphere. 
The magnetic globe is a spherical jigsaw. 

The MagnaCube is like a Soma (though not the same set of pieces), but each piece has a few magnets which constrain the solution. 
The Tricky Triangle requires you to position some spheres containing magnets so that they will not mutually repel. 
The pieces of the Blue Cube mate a certain way via magnets. Bits and Pieces also offers an aluminum version called "IsoCrate" by R.D. Rose. 

The object of Bumper Balls is to get the 3 balls separated 
Qbism 
OctaCube  8 subcubes attach to a central frame. Arrange the colors per rules. There is also a black nonmagnetic version. 



Pentera U.S. Patent 5411262  Smith 1995 
MagnaTease Classic Games Co. 
Laker Cubes 

(Mind Madness?) CubeIt 24 pieces  form a cube such that on the surface same colors don't touch (Saw one for sale here.) 
Mind Madness Ball similar to the graytoned ball above, but larger 
Balance of Power A dexterity puzzle or a game  the blocks have detents at various positions. Arrange the blocks in an attempt to place the magnetic marbles in the detents such that they remain stable. Score higher for using more closely spaced detents. The detent arrangement shown has maximal spacing. 

Dipole Dilemma by Chris Morgan Pack the 28 magnetic spheres flat in the rectangle 
Mattel Force Field 
Geometrix The Hexagon  Reiss Style 415 1980 

Olizoid See U.S. Design Patent D509263 awarded to Daniel R. Oakley in Sept. 2005. 
CHKD Cube Eight aluminum pieces with embedded magnets. Not difficult, but a child will be challenged. Based on the logo, this seems to be a promo item for the Children's Hospital of the King's Daughters. 
The Magic 16 ball by Idea Ocean, and the Tetrix Ball Twins from Mefferts. 

King Tut Magic Mummy Get the Mummy to stay in the Sarcophagus (or hop out) See U.S. Patent 2458970  Wilson 1949 
This is "Pack It In" from B&P, designed by Simon Nightingale  it is a onecube packing puzzle! The cube and box contain hidden magnets and a mechanism that prevents the cube from seating in the container until it is properly oriented and deliberately inserted. I include this puzzle here rather than in the "Packing" category since the magnets are integral to its operation. 
Here is a really nice version of Nightingale's One Piece Packing Puzzle, made by Eric Fuller in 2008. 

Tetraxis by KO Sticks LLC Produced with support from the Museum of Mathematics 
I decided to purchase one of John Devost's HexTwist Five Intersecting Tetrahedrons puzzles  it is a geometric artwork, a real dexterity assembly challenge, and a great example of John's woodturning skills  each rod is latheturned to have a special twisting shape. The rods are made from exotic woods, including: Purpleheart, Bloodwood, Redheart, Bubinga, and Lacewood. They are held together by magnetic tips and chrome balls. This puzzle has also been known as the "Poor Man's Merkaba" or PMM. 

Escherthemed Magnetic Cylinder Puzzle 
Balancing Act, designed and exchanged at IPP32 by Chris Morgan, made by Chris Morgan and Saul Bobroff 
A Kosticks puzzle a gift for IPP32 committee members. Thanks, Chris! 
Conundrums, Enigmas, Posers, Riddles, Quandaries, Rebuses, Catches, BrainTeasers...  there are many puzzles in the wider world beyond mechanical puzzles. I have added this section on logic puzzles so that I can document several puzzles that, while only occasionally presented as mechanical puzzles with physical pieces, have nevertheless provided some entertainment to me and my friends.
Many books over the decades have compiled logic puzzles. Among the most noted puzzle chroniclers of the Nineteenth Century are the Brits Henry Ernest Dudeney (18571930) (Amusements in Mathematics, 1917), Professor Hoffmann (penname of the Reverend Angelo John Lewis, 18391919) (Puzzles Old and New, 1893), and Walter William Rouse Ball (18501925) (Mathematical Recreations and Essays 1892), and the American Sam Loyd (18411911) (Cyclopedia of 5000 Puzzles 1914).
Modern chroniclers include Martin Gardner, Raymond Smullyan, Ivan Moscovich ( download and print some of Ivan's puzzles), Serhiy Grabarchuk, Ed Pegg, and Jim Loy.
Modern audiences still have a taste for logic puzzles, as evidenced by the current popularity of SuDoKu.
Online resources include:
Gridworks  Thinkfun 
Chocolate Fix  Thinkfun 
The Mensa "Challenge Your IQ Pack" contains several logic challenges. (Red version) 
The classic game of logical deduction Clue has been simplified and turned into a solitaire puzzle game called Clue Suspects. Given six rooms, a body, up to 11 suspects, and a challenge card providing a set of clues, deduce who must be in the room with the body, and therefore be the murderer. 
Logic Links By Mindware. Use given clues to deduce where to place colored chips. 
An old game show called "Let's Make A Deal" was hosted by Monty Hall. A contestant was allowed to choose one of three curtains. Behind one of the curtains was a valuable prize, while behind the other two lurked booby prizes. After the contestant made a first choice, the host would reveal a booby prize behind one of the unchosen curtains, and then give the contestant a chance to stick with their first choice, or switch to the other curtain not yet revealed, "where Carol Merrill is now standing." Either way, the contestant's final choice was then revealed, to either applause or laughter.
The puzzle asks, "Which is the better strategy for the contestant: always stick with one's first choice, or always switch?" This question caused quite a stir in the press when it was answered by Marilyn vos Savant, a noted columnist. Educated people wrote in to vehemently disagree with her answer, though it was correct.
I believe the correct strategy is easy to deduce  look at the two tables below. In each table, I show the outcome of one of the two strategies, based on all the combinations of where the prize is versus the contestant's first choice. Remember, it is always possible for the host to show a booby prize behind one of the unchosen curtains.
You stubbornly stick with your first choice no matter what the host reveals, so you win only when you picked the correct curtain on your first choice. Obviously your chances are 1 in 3. 
This is the better strategy. You lose only when you picked the correct curtain on your first choice. The host kindly eliminates one of the booby prizes for you, and by switching you end up winning 2 out of 3 times. This is counterintuitive enough to make people want to argue about its validity. 
You can pretend to be a contestant online here. The website tracks win/lose statistics, and they correlate well with the expectations noted above. Jim Loy also discusses the Monty Hall problem.
A friend told me this logic puzzle question was posed during a job interview. You are given two 60second fuses, and a lighter. Using only this equipment, time 45 seconds exactly. You cannot assume that the fuses burn at a steady rate throughout their lengths, only that they will each be completely consumed in exactly 60 seconds. You cannot cut the fuses  it would do you no good anyway, since by the previous statement there is no dependable correlation between any partial length and time.
Mouseover the box below to see the answer:
Light both ends of one of the fuses simultaneously, along with one end of the other fuse. When the first fuse (lit at both ends) has burned out, 30 seconds will have elapsed. Immediately light the other end of the second fuse, which will have already been burning for 30 seconds from one end. Its remaining length will now be consumed in 15 additional seconds. When it is gone, 45 seconds will have elapsed from when you first lit the fuses. 
Here is Dickinson's Witch Hazel Parking Puzzle, a vintage advertising card posing a logic problem. Five different cars must be arranged in a fivespace garage, from left to right according to a given set of constraints.
Here is a logical method of solving the Dickinson's Witch Hazel Parking Puzzle:
There are 120 ways to arrange the cars in the garage  you could put any of the five cars on the left, then any of the remaining four to its right, any of the remaining three to the right of that, etc  this gives 5x4x3x2x1 = 5! = 120 possible arrangements. However, the given conditions will preclude all but one of those arrangements. Can we deduce the proper arrangement without checking all 120 possibilities?
Each of the five cars must have something to both its left and its right  either a wall (if it is on an end), or one of the other four cars. Symbolize the Ford, Chevy, Plymouth, Packard, and Buick using the letters FCLAB, respectively, and an end using E.
Let's also state some assumptions about what is meant by LEFT and RIGHT, since the card doesn't explicitly define the terms  we'll park the cars facing INTO the garage, and we'll look at the puzzle from the view shown on the card  INTO the bays. So the LEFT side of the card corresponds to the LEFT side of a car, and the RIGHT side of the card corresponds to the RIGHT side of a car.
Construct the following chart:
Car on its LEFT on its RIGHT F EFCLAB EFCLAB C EFCLAB EFCLAB L EFCLAB EFCLAB A EFCLAB EFCLAB B EFCLAB EFCLAB 
At the outset, we don't know what is to the left or right of each car.
However, right away we can eliminate from each car's left and right lists that
car itself (unlike people, cars cannot be "beside themselves" :)
Car on its LEFT on its RIGHT F E.CLAB E.CLAB C EF.LAB EF.LAB L EFC.AB EFC.AB A EFCL.B EFCL.B B EFCLA. EFCLA. 
Now consider the first constraint, which gives us two facts
1. B and F cannot be neighbors, and 2. B and C cannot be neighbors. Eliminate the appropriate entries from the chart, replacing them with the numbers 1 and 2: Car on its LEFT on its RIGHT F E.CLA1 E.CLA1 C EF.LA2 EF.LA2 L EFC.AB EFC.AB A EFCL.B EFCL.B B E12LA. E12LA. 
Next, consider the second constraint, which again gives us two facts
3. L and C cannot be neighbors, and 4. L and A cannot be neighbors. Car on its LEFT on its RIGHT F E.CLA1 E.CLA1 C EF.3A2 EF.3A2 L EF3.4B EF3.4B A EFC4.B EFC4.B B E12LA. E12LA. 
The third constraint tells us 5. A and F cannot be neighbors 6. A and C cannot be neighbors Car on its LEFT on its RIGHT F E.CL51 E.CL51 C EF.362 EF.362 L EF3.4B EF3.4B A E564.B E564.B B E12LA. E12LA. 
Apply the last constraint, 7  eliminate L from the LEFT list of F:
Car on its LEFT on its RIGHT F E.C751 E.CL51 C EF.362 EF.362 L EF3.4B EF3.4B A E564.B E564.B B E12LA. E12LA.We'll use this final chart to solve the puzzle... 
Now we have to put the remaining possibilities together.
Consider the Chevy  row C.
On its left must be an end or the Ford, AND on its right must be an end or the Ford.
So, it HAS to be on an end, and the Ford must be next to it.
Therefore the arrangement looks like either:
CFxxx OR xxxFC.
Put this together with the info we have for the Ford  row F. On its LEFT can be an end or the Chevy. CFxxx satisfies this, but xxxFC does not, so we can discard xxxFC.
Now we can drive to a solution pretty quickly...
What can be to the RIGHT of the Ford? Its list says E.CL51  an end (impossible), the Chevy (impossible), leaving us with the Plymouth, giving CFLxx.
What can be to the RIGHT of L? EF3.4B tells us it must be the Buick, giving CFLBx, and the Packard must be on the right end (which is consistent with what the A row tells us  it has an end on one side and the Buick on the other).
Our final arrangement is CFLBA.
A Vanish Puzzle is a cleverly concocted illustration or geometric arrangement, showing a number of objects or a specific area and comprising pieces that, when rearranged, result in a seeming change in the number of objects or size of area depicted.
According to G. Frederickson in Dissections: Plane & Fancy,
the first example of vanishing area puzzles was reported in the book Libro d'Architettura Primo by
Sebastiano Serlio (14751554).
(Google books link)
The first description and mathematical explanation of the vanish paradox was found in a math puzzle book titled
Rational Recreations: In which the Principles of Numbers and Natural Philosophy are Clearly and Copiously Elucidated,
by a Series of Easy, Entertaining, Interesting Experiments. Among which are All Those Commonly Performed with the Cards,
by William Hooper 1774.
(Google books link)
According to Professor Douglas Rogers,
Hooper may have copied the puzzle from the book
Nouvelles récréations physiques et mathématiques
(Google books link)
by the French author Edmé Gilles Guyot (1770).
Martin Gardner discusses vanish puzzles in his 1958 book Mathematics, Magic and Mystery in chapters 7 and 8, and also in Wheels, Life, and Other Mathematical Amusements in chapter 12 "Advertising Premiums." Martin Gardner calls it "the principle of concealed distribution." There is an informative article by Mel Stover in the November 1980 issue of Games magazine.
L'Echiquier Fantastique is a French version of the geometric vanish Gardner called
Hooper's Paradox.

The Jerry Slocum collection at the Lilly Library contains a few examples of vanish puzzles.
Sam Loyd, the premier American Puzzlist of the 19th century, created a series of vanish puzzles, now classics, including Get Off the Earth. In one position, there are 13 Chinamen. Move the knob to rotate the inner disk, and one vanishes  now there are twelve! Patented and Copyrighted 1896 by Sam Loyd (563778). My copy was published as an art supplement to the Philadelphia Inquirer, Sunday July 12th 1898. Gardner informs us that more than 10 million copies were sold during Loyd's lifetime.
Click the image to make a Chinaman vanish!
Shown next is a 100th anniversary commemorative edition of Get Off the Earth. Sam Loyd reused this puzzle device in "The Lost Jap" and "Teddy and the Lions" (neither of which I have). These are discussed in Slocum and Botermans' "Puzzles Old & New" on page 144.
You can buy copies of Sam Loyd's vanish puzzles, and try them online here and here.
Bushwacky A modern version of Sam Loyd's classic Get Off the Earth vanish puzzle. 
The magician Theodore L. DeLand, Jr. (18731931) copyrighted a version of the vanish puzzle in 1907. It was printed in various forms, including "La Mysterieuse" (which I do not have). Gardner calls this the DeLand Paradox.
The Vanishing Leprechaun is another classic. It was designed by Ms. Pat (Patterson) Lyons circa 1968 and is related to the DeLand Paradox. Cut the card into three pieces  a long strip on the bottom, and the top into two sections. If you exchange the positions of the two top sections, there are 15 instead of 14 leprechauns. There is an explanation of the Vanishing Leprechaun at this site.
This is The Magic Egg Puzzle, copyright 1880 by Wemple & Company, New York. The card should be cut into four pieces along specific lines. The pieces can be rearranged in a rectangle so that 8, 9, or 10 eggs appear. The directions, however, also suggest that it is possible to arrange the pieces so that 6, 7, 11, or 12 eggs are shown.
Try the Magic Egg Puzzle online (requires Shockwave plugin).
That site also has an explanation of how these puzzles work. It also mentions a criminal use of the vanish: "The principle is very old and probably originated as a early method for counterfeiting money. William Hooper in his book Rational Recreations, published in 1774, described the paradox as 'Geometric Money.' It is possible to cut 9 bills into eighteen parts and then to rearrange them to make ten bills. To foil this method, the two numbers on all U.S. currency are placed on opposite ends, one high and one low. In this way, counterfeit bills using this method are easy to detect since their numbers will not match correctly. In fact, in 1968, a man in London was sentenced to eight years in prison for using this scheme on British fivepound notes."
I think it is interesting how Pat Lyons came out with the Leprechauns in 1968! Coincidence?
Here is a modern variant of the vanish, called Who Turned to Doggie Doo? by Robin Debreuil. You can see it on John Rausch's site here, and at Debreuil's site where you can download a free printable version. (Note: on Debreuil's blog, he put this in the public domain.)
Before:
After:
Geometricks is a beautiful small folio of five different and multifaceted dissection puzzles, copyright 1939 by M. Grumette, and published by EduKToy Institute, New York. It's in great shape for its age. Each page describing one of the puzzles is an envelope and encloses a card containing the corresponding puzzle pieces. The puzzle pieces are on good stock punchout cardboard  all of the pieces are present and intact, including the frames. I've tried to show a glimpse of each page/puzzle below.
Copyright 1939!  
VersaTiles

'Teen Squares The four pieces can be arranged to apparently show a total of 15, 16, or 17 black squares. A classic geometric vanish. 
Biform Square

The Tormenter

HaCho Form various silhouettes from the seven tiles. 

Here are some examples of 3D Geometric Vanish Puzzles, which are related to Packing Puzzles.
HABA Trickpack See my solution below. 
Conway Packing Puzzle A gift from Brett. Eq. to HABA TrickPack. 
Think Square  Pressman There are 4 small right triangles, 4 large right triangles, 4 staircase shaped pieces, and 5 small squares. The pieces can be fit snugly into the tray with and without one of the five small squares. 


You can find many other examples of vanish puzzles on the web:
Puzzles based on a picture can be printed on paper or card stock. Some are to be cut up and arranged in a particular way. Some call for you to find various figures in the pictures.
This is a French puzzle called Un Sage Dans Les Nuages  "A Sage in the Clouds." Four rectangular cards depict various cloudscapes. Arrange them so that the face of a wise old sage appears. I don't think the face is very wellformed. Shown in Slocum and Botermans' "New Book of Puzzles" (1992) on page 23. 
Another French puzzle, called Les Quatre Vagabonds  The Four Vagabonds. Arrange the four cards to form one complete figure. Appears in Hoffmann as Chapter III No. XLI  The Man of Many Parts. Hoffmann says it is of German origin. 
L'Astronome  arrange the three pieces to form a fivepointed star, with an image of the astronomer. 
Quelques Tours dans une Boite  In addition to the loose versions of the above puzzles, I obtained this boxed set which includes the four paper puzzles Les Quatre Vagabonds, Un Sage dans les Nuages, L'Astronome, and L'Incroyable (a paper version of the geometric fallacy L'Echiquier Fantastique). You can find an online version of this set here, with links to cards you can print and cut out. 
Mystery Picture Something New and Novel "Lindy" Look steadily at small diamond shaped speck on nose try not to blink and count to 50 slow, then look up at the sky day or night or on a light wall and photograph will appear greatly enlarged. Keep looking at one spot for 10 seconds. Result  The actual photograph will appear and disappear several times. New! Startling! Amazing! 
One classic, popularized by Sam Loyd, is seat the riders on (or saddle) the horses (or mules). See U.S. Patent 2082943  Dutcher 1937. Cut out the three pieces and figure out how to arrange them to depict two complete horses each bearing a rider facing the correct way.
This puzzle was used as an advertising premium for Dickinson's Witch Hazel  seat the witches on the cats:

Find: a Queen, Lady, Traveller, Hostler, Clown, Boy, Baby, Gorilla, Monkey, 2 Donkeys, 2 Horses, Elephant, Bear, Deer, 2 Rabbits, 2 Squirrels, 3 Frogs, 5 Dogs, Otter, 2 Turtles, 10 Faces, 29 Letters, Bird, Rat, 2 Fish, Owl, &c (That's what the card says, "&c"  I guess they got tired of listing the items!)
Find: a Bear, Buffalo, Camel, Giraffe, Seal, Swan, Squirrel, Cat, Fox, Pig, Rabbit, Parrot, 2 Alligators, 4 Birds, 2 Beavers, 2 Babies, 2 Boys, 5 Cows, 2 Chickens, 2 Deer, 12 Dogs, 3 Elephants, 3 Frogs, 3 Fish, 7 Faces, 2 Goats, 7 Horses, 10 Letters, 2 Mice, 4 Men, 2 Monkeys, 2 Owls, 3 Rats, 3 Sheep, 2 Turtles, 2 Ladies.
Find: an Elk (not Miss Anne Elk :), Peacock, Shark, Butterfly, Lion, Tiger, Rabbit, Book, Coat, Boot, Hare, Rake, Barrel, Caterpillar, Pigeon, Yard Stick, Snail, Match, Turtle, Owl, Rhino, Antelope, Watch, Skull, Cat, Cow, Giraffe, Priest, Mummy, Humpty Dumpty, Squirrel, 5 Fishes, 2 Indians, 12 Faces, 3 Mice, 11 Dogs, 3 Eagles, 5 Letters, 5 Ducks, 2 Camels, 3 Elephants, 7 Men, 2 Monkeys, 2 Cymbals, 4 Birds, 4 Bears, 4 Goats, 8 Frogs, 2 Seals, 3 Beavers, 9 Sheep, 3 Ladies, 5 Horses, 5 Pigs, 2 Chickens, 4 Alligators, 2 Boys, 2 Babies, and 2 Combs. Whew!
The Truant Boys, a followup to the Toll Gate series, by the same Dr. Abbey:
If you have Slocum and Botermans' "New Book of Puzzles" (1992) you can find several paperbased puzzles...
Here are the "cube snakes"  unit cubes linked together so that they pivot in only certain ways, and fold up into a cube shape, usually 3x3x3 but some 4x4x4. In this group are:
Another type of folding puzzle is the "plate" puzzle exemplified by Rubik's Magic. A group of independent 2sided grooved tiles, with embedded picture cards, are connected by a loop of strong fishing line. The stringing pattern is complex, and permits the tiles to be folded around and onto each other in various ways. The objective is to achieve a particular picture pattern and/or shape.
Take a look at Pantazis' site  he has created many original folding plate puzzle designs!
Rubik's Magic  the original 8tile, black verison. Newer versions are red. Read about Rubik's Magic at Jurgen Koeller's site. 
Rubik's Magic Master 12 tiles 
Rubik's Magic Create the Cube 
Simpsons novelty Rubik's Magic 
Another novelty version picturing a scantilyclad woman. 
Betcha Can't is a fairly rare version with hexagonal plates. 
Magic Smile  Purchased at IPP28 in Prague, from Pantazis Houlis also Mr. Twisty, designed, made, and exchanged at IPP32 by Pantazis 

I got this custom 4tile magic from Juozas Granskas at IPP26. 
Custom 8tile Magic  Prague Purchased at IPP28 in Prague, from Juozas Granskas 
A custom 3tile Magic, designed by Juozas Granskas. There are three nicelydrawn whimsical characters, each divided across two tile surfaces. The objective is to match the heads to the bodies so that all three characters are whole, simultaneously! I really like this puzzle, and not just because I can solve it. A gift from Jouzas at IPP 29 in SF  thanks! 
This is the "Pick the Pickaninnies" postcard puzzle, patented June 4, 1907 ( U.S. Patent 856196  Lehman 1907  CCL/273/155) and copyright 1907 by the Ullman Manufacturing Co. of New York.
This puzzle is representative of a time in the history of the United States during which what are today unconscionable racist sentiments were part of everyday life.
I have a copy sporting a canceled one cent Ben Franklin stamp, postmarked Feb. 4 1908 out of Philadelphia. The puzzle consists of a single card with six flaps that will fold over a central rectangle. One side of the central rectangle is the face of the postcard. On its interior face is an advertisement for Harry H. Kurtz furniture of Philadelphia. One flap depicts a black woman who is saying, "Show me all dem eleben pickaninnies at one time. I don't want to see no white trash." This flap also has eight holes and three black children's faces on it. The other flaps have various patterns of holes and faces of black children and white children. The objective as stated on the card is to "Arrange the flaps, by placing one over another, in such a manner as to show only the eleven pickaninnies."
This style of puzzle has appeared in less offensive versions, for instance the face of Danny Kaye was used to advertise his film "The Inspector General" and a can of peas. In that version only seven faces must appear. Below is a version I made for you to cut out and try. Make windows wherever it says "cut out" and remember to separate the side flaps.
Power Puzzles is a set of ten colorful folding puzzles made of heavy plasticized stock. The designs are copyright by the IvanConcept Corporation (Ivan Moscovich), and the set was distributed by Discovery Toys Inc. Each puzzle is to be folded to achieve some specific pattern. 
This is Fold A Decathlon of Mindbending Folding Puzzles, by Ivan Moscovich, issued by the Fatbrain Toy Co. It is a reissue of the Power Puzzles set put out by Discovery Toys. 
Fold the sheet to find the fifth pig...
Proper folding of the sheet produces a drawing of Hitler.
This style of puzzle has been used more recently to challenge you to "Find the Fifth Dinosaur"  Saddam Hussein.
Fold a camel. The starshaped sheet is printed on both sides and has a slit to the center cut along one radius. See U.S. Patent 2327876  Edborg 1943.
In this puzzle, you fold one way for a cow, another for a horse: U.S. Patent 2327875  Edborg 1943.
Rubik's Snake is not so much a puzzle as a plaything. Twist it into different shapes. 
Here is a folding metal "puzzle" (more of a toy) known as "Heaven's Orb." This design actually dates back at least to the Smith patent # 2031231 of Feb. 1936. 
Triamant also sold as "Crazee Diamond" 
CanDo Linked blocks in the shape of a can. See U.S. Patent 6637138  Prost 2003 
Yoshi's Cube A flexible array of shapes connected by tough plastic sheets. Can be folded into a cube and other shapes. 
Shinsei Twin Comets or Shinsei Mystery Two units  either can be folded to make the first stellation of the rhombic dodecahedron, or a cube. Each shape is hollow  the stellation will fit inside the cube. This is a copy of a version called the Yoshimoto Cube that was first issued in Silver and Gold colors. Invented by Naoki Yoshimoto in 1971. 

Inca  small and large 
Rubik's Maze 

This is Hex, issued back when Thinkfun was Binary Arts. Fold the chain loop into a hexagon so that all arrows face the same direction. A gift from Tom Jolly  I had an instruction sheet but no puzzle  thanks, Tom! I've seen other versions of this on Iwase's site. 
A plastic foldingplate "puzzle"  make different shapes. 

Happy Cubes by Adult Games. 
BlockAids Identical to Happy Cubes by Adult Games  just blue. 

Snoop Cube Fold the eight linked blocks into various shapes, culminating in a cube. Purchased from Torito. 
MagNif Curious Cubes 1982 
The Starbix folding puzzle/toy, by Alan H. Schoen and issued by Bandai 1987. 

Cubigami 7 from George Miller. Designed by George Miller and Donald Knuth. This is one of the few puzzles that stays on my desk  I find myself frequently picking it up and playing with it. A clever arrangement of hinged squares in a flat sheet, can be folded into each of the 3D tetracube shapes except for the 1x2x2 block. 
Blue Cubigami from George Miller. This version has four magnetic blocks and a plastic wrapping with embedded metal plates. Arrange the four blocks into one of the tetracube shapes then find a way to wrap them. This version allows the 1x2x2 shape to be wrapped since leaves of the wrapping can overlap. A gift from George  thanks! 
Der Umstulpbare Wurfel (the Invertible Cube) by Paul Schatz of Switzerland. 

This is called "Block Chain" but it is a copy of QRIN X by Takeyuki Endo 
Two vintage folding puzzles from Binary Arts: GeoLoop and GeoMorph12 
Mind Jewel, designed by Alexander Polonsky  from RecentToys 

Betty's Baffling Bracelet  designed by Stuart Gee Made by Brian Young Exchanged at IPP27 by Marti Reis Six octahedral pieces strung on an elastic band  fold them to create a selfsupporting rhombic dodecahedron. 
Tony's Hinge 

Unhinged by Thinkfun 
The Not So Strait of Dover by Scott Elliott Kind of folding, kind of tangling  braid the strands using a classic "trick." 
This is Ivan's Hinge, another design by Ivan Moscovich, also issued by the Fatbrain Toy Co. 

Hopson Kinetic Prismatoy 
Brainwright has issued several intriguing new puzzles.
I picked up the FlexiCube, designed by George Miller. 
Manifold  issued by Brainwright A pad of 9cm^{2} sheets, printed with various patterns of light and dark areas. Fold every sheet so one side is all light and the other all dark. Gets quite tricky! Manifold was developed by Jérôme MorinDrouin at The Incredible Company. You can download a PDF with five sample challenges at their website. 
Here one has to make a careful distinction between games and puzzles. I believe all of these qualify as puzzles...
My first and still my favorite electronic puzzle was a gift from Darcy, the XL25. In principle it is very close to the later Lights Out  but IMHO it has the best sound effect and the best coordination of that sound effect with the push of the buttons. 
Shifty  Tiger 1989 (Read more at the Handheld Games Museum.) 
Tiger has marketed several versions of Lights Out. You can also get a version for your Palm PDA. 
Luminations rather than button pushes it requires tilting 
Nemesis Factor 
Orbix 
Genius Milton Bradley 1980 
Merlin Milton Bradley 
Lite 3  Tiger 
Rubik's Revolution 
The Cubed Electronic Puzzle, from ThinkGeek. 
Upside 
The Rubik's Touch Cube (debuted at $150, purchased for $50 at Best Buy. I think it eventually fell to $19.99!) 
Rubik's Slide See a short video review of the Rubik's Slide. 
Cool Circuits 
1Bit Puzzle by Zach Radding An enigmatic 1" square circuit board with a red LED in the center and a membrane button on each of the four top edges  marked 1, 10, 100, 1000. Insert the coin cell battery and the LED starts blinking. What does it all mean? Voted one of the best puzzle finds of 2012. Thanks, Zach! 
Laser Maze  produced by Thinkfun and

This is a classic  the Chinese Finger Trap. Insert an index finger into each end, then try to extract them! The woven tube grips more the harder one pulls. 
This is known as a Rattleback or a Celt. Due to its peculiar crosssection, no matter whether it is initially spun clockwise or counterclockwise it always ends up spinning in one direction. 
This "Magic Ring" trick consists of a fairly long loop of chain and a metal ring. Hold the chain in one hand and pass the chain through the ring. Now drop the ring, and if you know the technique, the ring will "magically" knot itself onto the chain! 

From James Dalgety  the Rabbiduck. Is this a Rabbit or a Duck? Don't answer until you click on the image... The Rabbit/Duck ambiguous image was published by Joseph Jastrow in 1899 in Popular Science Monthly  he discovered it in an 1892 Harper's Weekly, which had reprinted it from the German magazine Die Fliegende Blatter. You can read an article on the Jastrow Illusion, Joseph Jastrow and His Duck  Or Is It a Rabbit?, by John F. Kihlstrom, online. 
Bottomfilling Teapot and Fairness Cup set Purchased at the 2011 New York Puzzle Party hosted by Tom Cutrofello, in Manhattan at the Museum of Chinese in America (MOCA). MOCA was housing the exhibition "Chinese Puzzles: Games for the Hands and Mind" curated by Wei Zhang and Peter Rasmussen. These are examples of Puzzle Vessels. In the case of the teapot, it is filled through a hole in the middle of the bottom, yet when it is righted it does not leak, and pours tea upright as normally. The Fairness Cup, also known as a Greedy Cup, or Cup of Pythagoras, can be filled and used as normal, but if it is overfilled past a certain level, all the liquid will drain out through a hole in the bottom. The puzzle is, how do they work? 
This a modern example of a Chinese Magic Mirror. When sunlight reflects off of its polished brass face (opposite to that shown) properly, one can see an image of the IPP Burr puzzle logo. The unique production technique was invented some 2000 years ago in China and entails more than ten complicated procedures. The reflective surface has extremely subtle warps. The second photo captures the image projected by my copy. 

Vintage promotional puzzle Which is Larger? 
Fritz and Paul A version of the "Which is Larger" optical puzzle made in Germany "New! Great! Amazing! The droll Piccolos or the enigmatic twins Wins every bet very amusing!" D.R.G.M. Reg. i. a. K.  Staat Franz Wieland, Berlin S. 59, Camphausenstr. 25. 
A while back I received from Scott Elliott (Thanks, Scott!), a copy of his Screwy Screw  an "impossible object" type puzzle where the objective is to figure out "how did he do that?" In this case, the two nuts spin on or off the bolt in opposite directions! I.e., one spins clockwise to go on and the other spins counterclockwise to go on. Scott discusses this puzzle on his blog, here, here, here, and here. 

A beautiful handturned OffsetSpinningTop, made by Stephen Chin The puzzle is, how was this lathed? And, when it spins, it displays the intriguing effect of appearing to have independent disks hovering in air. Thanks, Chinny! 
Promotional puzzle from IBM What solid shape will fit through each hole, completely filling the outline? Appears in Wyatt's 1928 "Puzzles in Wood" as "The Wedge Plug Puzzle." Also issued by Journet as "The Geometrical Puzzle." 
Lenz's Law Demonstration
A moving magnetic field induces an electric current in a nearby conductor  this is called an eddy current. In turn, the induced current will create its own magnetic field that opposes the original magnetic field that created it.
One can demonstrate this using a simple copper tube and a strong magnet that will drop through the tube. The copper tube conducts but is nonmagnetic. The magnet will drop through the tube, but much more slowly than a nonmagnetic equivalent mass. It's not "sticking" to the tube  rather, Lenz's Law  the opposing magnetic fields  slows it down.
A gift from Brett  thanks! 

According to Jim: "This locket transformed from an oval into a heart shape and opened to mysteriously reveal a picture. There was no apparent way the locket could function in this way without tearing the picture in half." The puzzle here is "How does it work?" You might be able to find one on eBay. (There are several makers, I think Jim's is of the highest quality.)

A bank containing a suspended cube. Where does the money go? 
A bank within a bank. 
An impossible nail in a donut. 
An impossible Penny in a Bottle 

CardFolding Wizardry by Louis Coolen  a gift from Allard Walker. Thanks! It was great meeting Allard and several other puzzlers at IPP32 for the first time in person. I wish I had had more time to socialize. 
Here are some patents to early impossible object puzzles:
There have been many World's Fairs held since 1756. I have included only a small number of the betterknown Fairs in the table below. Some puzzles are among the memorabilia available from the various Fairs. Some of these items are shown on other pages, but I thought it would be interesting to assemble them here, too.
1851 Great Exhibition London, England 
The "Great Exhibition of the Works of Industry of All Nations"
(or just
"The Great Exhibition"
for short)
was based on an idea of Prince Albert's, was held in 1851 in the Crystal Palace in Hyde Park, London, and
was the first international exhibition of manufactured products.
The Crystal Palace is the icon of this fair.
I believe this vintage jigsaw puzzle is a souvenir. It is 6" x 6" x 1/4" thick, and the lid is marked: "The New Puzzle. Registered Industrial Exhibition  Registered According to Act of Parliament" There is a label inside the lid, probably from a store: "Rich d. S. Williams  No. 9 Back of Park Street  Bristol"


1876 Centennial Exposition Philadelphia, PA (Take a tour!) Or, read a book about the expo. 
A set of jigsaws (I don't have this). 

1889 Exposition Universelle Paris, France 
This fair featured the Eiffel Tower, but I haven't run across any puzzles.  
1893 The World's Columbian Exposition Chicago, IL 
The 1893 Columbian Exposition, by all accounts, was a fantastic event.
The buildings were clad in white plaster and the grounds were known as the "White City."
Peter Nepstad has created a
text adventure game
that puts you at the Fair.
Read
some
reviews.
Download a demo.
Also see
The Devil in the White City
by Erik Larson.
There were several other puzzles sold in conjunction with the fair:


1901 PanAmerican Exposition Buffalo, NY 
The Stars and Crescent puzzle was stamped to commemorate the 1901 Pan American Expo in Buffalo NY. I have this puzzle, but it's not stamped. 

1904 The Louisiana Purchase Exposition St. Louis, Missouri 
I have the Key, the UpToYou, and the sliding piece puzzle. I don't have the Jug or the furniture jigsaw.
Here is another interesting puzzle from the Expo  it's a pair of coins with slots in them. I obtained a set including the strap! I found a U.S. patent describing the puzzle and the method of [un]linking them with the strap: 748245  Willey & Barton 1903


1915 PanamaPacific International Exposition San Francisco, CA 

1933 Century of Progress Chicago, IL 
There were several puzzles associated with this fair.


1939 New York World's Fair NY, NY 
The Trylon Perisphere is the icon for this fair. A small interlocking puzzle of the Trylon Perisphere was issued and became the forerunner of the keychain puzzle. There was also a version in metal  I have the plastic version.


1962 Century 21 Exposition Seattle, WA 
I don't have this. 

1964,65 New York World's Fair NY, NY (NOTE: Not sanctioned by the BIE.) 
I don't have this. 

1967 Expo '67 Montreal, Canada 
Here is a brief history of "ancient" puzzles, and of "modern" puzzle crazes.
What is the oldest mechanical puzzle? We shall probably never really know  after all, the object itself or a record of it would have to have been preserved, found, and accurately dated. Only objects made of robust material, such as ceramics, are likely to have survived their trip down through the ages. And even if an object were found, without some written or pictographic record of its intended use, we can only surmise that it served the purpose of a mechanical puzzle. That said, below are some candidates (all dates are, of course, approximate). I have included some mathematical, logical, and word puzzles, too. (Jerry Slocum made several helpful comments in private correspondance.) David Singmaster has undertaken a far more detailed Chronology of Recreational Mathematics.
ROTAS OPERA TENET AREPO SATOR 
See a timeline of games and puzzles here.
Meffert hosts a timeline of puzzles written by Prof. David Joyner here.
See Puzzlehistory.com (focused on jigsaws).
Read a history of mathematical games and recreations here.
Check the Usenet rec.puzzles archive.
See a large collection of images of word games here.
See the MSN Encarta entry on Puzzles here.
Dictionary.com defines "craze" as "a popular or widespread fad, fashion, etc." and as a verb, "to cause to become mentally deranged or obsessed." As can be seen from the history of ancient puzzles above, people through recorded history  even workmen building the pyramids  have enjoyed puzzles. Many puzzles have made it into the historic record  often recorded by and for an intellectual minority  those who could read and write. However, few puzzles seem to have created the kind of fervor that causes masses of people to lose sleep and ignore their obligations, thereby instigating a fullfledged craze.
I believe it is not surprising that crazes seem to be a relatively modern phenomenon  it is only recently that there exists a mass population with sufficient leisure time to devote to "frivolous" pursuits, manufacturing capability sufficient to produce enough copies of a puzzle, transportation speedy enough to spread the puzzle widely within a relatively short period of time, and communications media sophisticated enough to report on the craze while it is occurring, often intensifying the craze by exposing more people to the puzzle. Records must survive that document such widespread popularity and, hopefully, chronicle the craze.
Here are some puzzles for which records do exist  the exceptional manias are highlighted, and some lesser crazes are included, too:
Since 2001, the IPP has held a design competition, now known as the Nob Yoshigahara Puzzle Design Competition. You can see the entries and results at John Rausch's site.
Throughout my website, you'll find many other puzzles that have appeared in the Design Competition in recent years  below, I've gone through the entry lists and noted the puzzles I've acquired:
2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

Several manufacturers around the world, from the late 1700's onwards to the present day, have issued collections of puzzles in boxed sets.  
Perhaps the most soughtafter are collections of puzzles made from ivory, arranged inside lacquered boxes, exported from China in the 1800's. One such set is shown here  sadly I do not own it  it sold for over $4000! The HordernDalgety Puzzle Museum site has an article on Chinese Puzzle Sets. There is also an article on the British firm John Jacques & Son, who made boxed sets of puzzles, indoor and outdoor games, and other items. Jacques of London was founded in 1795 and is still around today! 
Other boxed sets of puzzles have come from French manufacturers, such as the "Jeux Nouveaux" set shown above
(I do not own).

This boxed set contains a Soma cube, a Star, and a Cube Snake. 
The "Aha Brainteaser Classics" set from Thinkfun contains a nice introductory survey of simple mechanical puzzles, with hint cards. Here is a nice history of some of the puzzles included in this set. 
This compendium is called "Mixed Up." 

This compendium was made by Sherms of Bridgeport CT (the included instruction sheet clearly indicates so, as do the several appearances of the leering devil), but branded on the box lid by Kellogg's of Springfield MA (probably a department store). Although the lid says "Wire Puzzles," the set contains the Perigal 4piece square, the classic T dissection, Loyd's buttonhole pencil, the 8point puzzle, a sitting Doggie puzzle, several metal tanglements, and a 14piece checkerboard dissection called the "Checker Board Problem" made from thick cardboard in blue and red (Haubrich 14.14.35, pieces on page 165, listing on page 168 #11). The instruction sheet says "This book of puzzles explains others besides those in your set." It's not at all clear whether the current contents are the original contents. The instructions sheet lists:

This is the "Party Puzzle Box Supreme," a compendium of several puzzles, from George E. Schweig & Son of Philadelphia PA. The box is in poor shape, but it contains several interesting puzzles. The box contains three trays / levels. The first tray contains four sliding block puzzles in good shape: Schweig's TransAtlantic Puzzle, The Flying Puzzle, The Traffic Jam Puzzle, and Ma's Puzzle. (I bought this to obtain the TransAtlantic puzzle.) The next tray contains eight hard wire tangles, some of which are incomplete, and a dexterity puzzle called "Who Catches Us?" The last layer contains the classic T dissection (missing one small part), "The Wonder Puzzle" (a tangram variant, missing a block or two and a booklet), a "Spoophem" type puzzle, six more tangles including Patience, and a sixpiece burr, the same "Puzzle of Puzzles" as in the "Tricks and Puzzles" compendium. There is also an instruction sheet. 
This set is called "Tricks and Puzzles for Young and Old" and I cannot find additional provenance info even on the instruction sheet. One of the included tricks, however, called "Mystifying Mind Reading," is labeled No. 1 N 136 Copyright 1928 N. S. Co. Chicago. Also, many of the individual item packages say "Made in Japan." The set includes: the aforementioned Mystifying Mind Reading trick, which contains six cards having pictures of 15 presidents; a traditional sixpiece burr puzzle called "The Puzzle of Puzzles" made in Japan (pieces 1, 154, 256x2, 1024x2), four copies of the ring and clip hard wire tangle, two copies of a hard wire tangle similar to offset keys, plate metal horseshoes, two copies of "No. X 2234 Coin or Disc Thru Hole," "The Magic Sex Indicator" (a plumb bob), an "XRay" tube, a card trick (incomplete), and a sheet of instructions. The instructions describe:


De Luxe Puzzle Set  issued by D. Robbins & Co. N.Y.C Copyrighted 1953 A nice set of four classic brainteasers. I picked up spares because the first 3 of these are popular items in my "puzzle go bag." 
Screwy Balls Get 3 black balls on one end and 3 red on the other. A secret trick is involved. 
Puzzle Ring 
a version of "Upsy Downsy"  get the ball up the ramp to the pinnacle 
TileORama 
Magic H 
I'm not sure what might be missing, but the following items were also included:
Open my Virtual Mechanical Puzzle Compendium Box to try various online simulations of mechanical puzzles.