Bill Cutler Puzzles, Inc.
Practically everyone with whom I have communicated has had
something to say about my use of the word `level.' I agree that the
results in the computer output is in many cases smaller than what most
people would count as `moves,' which would be used as an estimate of
difficulty. The use of `level' is tied directly to the method of
disassembly analysis that BURR6 employs. It relates to the tree depth
at the point where the first solution was found. Each new level of
the tree involves another call to the major subroutine in the program
which analyzes all movement in one direction. I would be happy to try
and improve the use of this word, but there are some basic questions
which need to be answered. The following examples should make clear
some of the problems:
- Everyone will probably agree that Bill's Baffling Burr (BBB) has a
level-5, or 5-move solution. Each of the first four moves
consists of moving a single piece in one direction one unit in
length. The fifth and last move results in separation of one
piece from the rest.
- Most everyone will also agree that the first level movement in
Bruce Love #1, Assembly #14, should count as two moves, or
levels. The movement listed is for one piece to move one unit
in length; a second piece to move one unit in the opposite
direction; and the other four pieces to remain in the same place.
Furthermore, the next level movement requires that both of
these pieces be moved, and an analysis of the list of states
reveals that the only solution lies with these first moves, so
what the computer lists as a level-5 solution should be called
a 6-move solution. (Note that in Bruce Love #1, Assembly #13,
a similar first move does not need to be done in the same
fashion, with the movement for piece #6 being added to the
level-3 movement. However, the same situation now exists with
the level-3 movement.)
Now, for the difficult case -
- The fourth move in the Gaby Games (GG) burr: In the movement as
listed in the output, two pieces are moved the same amount in
the same direction. They can also be moved separately, one
first, and then the other. In fact, most people, when
physically manipulating the puzzle, will make these moves
separately. How many `moves' is this? One-and-one-half?
How `difficult' is this move? Certainly more difficult than a
move from item 1 above, but does it rate the difficulty of item 2
And more -
- What about the three moves in Stewart Coffin's Triple Slide? Each
of these moves consists of three of the pieces moving as a unit in
relation to the other three. Is each of these moves more
difficult than one of the single moves in BBB? Are these
moves as hard as the fourth move of GG (by computer's output)?
And still more -
- What about the first move in Bruce Love's B-4 design? The
move is a single piece of only one unit in length, but the piece
is able to move two units, and the second move can only be made
if the first move is exactly one unit, so that a `lock-picking'
type of movement is made. The first move would surely count
as only one move, but is clearly more difficult than other
My own choice for an estimate of difficulty would focus on the
number of `states' that might be encountered prior to disassembly. At
this time, my best suggestion is to count all states at a level less
than the solution level. This might be modified to (a) add states at
the solution level which do not lead on the next step to a solution,
or (b) remove states at the level below the solution level which lead
on the next step to a solution. In any case, such a count would
include `false' moves or `blind alleys' and multiple ways of arriving
at a solution; both of which may cause the solver some difficulty.
I await your suggestions. The next version of BURR6 can be
changed to produce output that more closely approximates the idea of a
`move,' but it must first be decided how to handle the above cases.
The following comments are concerned with the difficulty in
implementing changes to the program:
- The program produces a solution with the minimal level
(subject to its interpretation of `level'). It does not do any
comparison of solutions of the same level in order to find the
- Revising the program to list moves such as Bruce Love #1
first move as two separate moves is not too difficult to do.
- The program makes no attempt, when disassembling the
initial assembly, to take into consideration how difficult some subassemblies
may be to disassemble. The program minimizes only the level of solution of a
particular assembly or subassembly. Assemblies and subassemblies are all
treated as separate problems as far as looking for the lowest level.
Including code to minimize `level' across the subassemblies that follow may be
Copyright 2000-2002 Bill Cutler Puzzles, Inc.