What should 'Level' be? billcutlerpuzzles.com
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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:

1. 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.

2. 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 -

3. 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 above?

And more -

4. 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 -

5. 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 single moves.

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 `simplest.'
• 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 quite difficult.

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