Archive for April, 2014

A Magic Dart Puzzle

April 16th, 2014

Recently I was looking at the following figure, in the context of trying to find useful configurations to make crossed stick puzzles with:

golden-dart

Now, a four piece crossed stick puzzle seems likely to be rather trivial to solve, but it sparked a question: could there be a good, non-trivial puzzle that uses this dart configuration? I recalled that I had in the past made such a figure with craft popsicle sticks:

magic-dart-blank

This assemblage of sticks suggests some sort of puzzle where there will be some figures printed on the sticks at the intersections between the pieces, and there will be some goal concerning the figures that are showing on the sticks that are on top at the intersections. With four sticks, there are 4! permutations of the sticks, and if each stick can be flipped over or rotated 180°, there are 4!·44 = 6144 possible arrangements of the sticks. (Because we can flip over the entire assembly, there are only half as many physical arrangements of the sticks, but it seems reasonable to consider either side of an assembly to represent a different arrangement, since we will probably want a goal for the puzzle that is only concerned with what is visible on one side.)

In the crossed stick puzzles that I have looked at thus far, the only sites that vary have been the intersections between pieces. However, if we allow rotating the pieces, there will be sites that can either be at intersections or not, depending on how the piece is rotated.

magic-dart-sites

I first toyed with using colored marks at the variable sites, possibly with holes in the pieces that could reveal the color of the piece behind it. However, I made more progress when I turned to the idea of putting numbers at the sites. Since there are eight sites on the two sides of a piece, I thought of putting each number between 1 and 8 on one of them. This suggested to me a puzzle along the lines of a magic square, where the goal would be to make a figure where the sums along the four lines all matched. I noticed that there are exactly four ways to partition the numbers 1 to 8 into two sets of four, such that each set has the same sum (18):

1 2 7 8 | 3 4 5 6 
1 3 6 8 | 2 4 5 7 
1 4 5 8 | 2 3 6 7
1 4 6 7 | 2 3 5 8

That would give me an elegant way to place numbers on each side of the four pieces, but I still needed to find a way to arrange the numbers. The simplest way would be to place them in order from smallest to largest. I wrote a short Python program to output the solutions. Unfortunately, there weren’t any! After that I tried other orderings. (Note: due to a bug in my program, what I previously had here was incorrect.) If we number the sites from smallest to largest as 0 through 3, ordering the sites as [0, 2, 3, 1] gives a single unigue solution:

magic-dart

Ordering the sites as [0, 3, 1, 2] gives four solutions with a magic sum of 18, and four solutions with a magic sum of 17. Notice that magic figures like this have a numerical “reflection” symmetry. Given a solution, you can generally obtain another by replacing every numeral n with 9 – n. This means that the orderings [2, 3, 0, 1] and [3, 0, 2, 1] will behave basically like the orderings mentioned above. Other distinct orderings with solutions are [0, 1, 3, 2] (7 solutions), and [1, 0, 3, 2] (2 solutions).

I’ve only started looking into how I could physically produce such a puzzle. Custom printed and die cut PVC looks promising; I think the marginal cost of a piece would come to under 10¢; the real problem would be in the setup cost, and in the unlikelihood of minimum orders of less than 500 for each piece being feasable.

As an aside, once I was looking at those partitions of the integers 1-8 into eight subsets with equal sums, the obvious thing to do with them was to make a 4×4 “magic square” where each of the subsets occurs once as a row or column. Here’s one where the diagonals also have a sum of 18:

magic square of partitioned subsets

This seems like the sort of idea that would have to have been looked at before, but I have no idea how to find out where it may have been previously discussed.