Logo maze puzzle: analysis and solution by Gordon Collins
 Gordon Collins, puzzle solver extraordinaire, sent us the diagram above with the three subcircuits highlighted in different colors. He then transformed this pattern into a stretched string figure, below, by grabbing the two balls and pulling them apart. Numbers with asterisks are the balls. From this pattern Gordon was then able to deduce a minimum of 14 and a maximum of 64 moves! Here are some of his sample solutions: The smallest number of moves is 14: 1e, 1f, 1m, 1k, 1n, 2e, 2a, 2d, 2h, 2i, 3a, 3b, 3l, 3*. One can use 2f and 2b instead of 2e and 2a. There are many ways to get there in 25 moves, one of which is: 1b, 1h, 1i, 1d, 1c, 1l, 1k, 1n, 2e, 2j, 2k, 2b, 2d, 2h, 2g, 2l, 2m, 3k, 3j, 3e, 3f, 3d, 3c, 3l, 3*. For the maximum, there are several choices for restrictions on the problem: cannot return to start or arrive at a given end point more than once; cannot arrive at a given endpoint from the same direction more than once (prohibits following both 2f2b and 2k2b); cannot use the same directed path more than once (prohibits following 2k2b twice); cannot use the same UNdirected full segment more than once (prohibits following 1m1d and 1d1m); cannot use the same undirected full segment more than once in succession (allows following 1m1d and 1d1m, just not one right after the other); and cannot use the same directed full segment more than once in all. Gordon found the following maximal paths for the above: 35 moves: 1b, 1h, 1g, 1i, 1m, 1d, 1c, 1l, 1k, 1a, 1n, [ or ... i d c l m k ...] 2f, 2e, 2a, 2j, 2k, 2b, 2d, 2h, 2i, 2g, 2c, 2l, 2m, [2 other variations before d] 3a, 3k, 3j, 3e, 3f, 3g, 3d, 3c, 3b, 3l, 3* 52 moves: (reaches all endpoints at least once) 1e, 1f, 1e, 1*, 1b, 1h, 1g, 1j, 1g, 1i, 1d, 1c, 1l, 1c, 1d, 1m, 1k, 1a, 1n, 2e, 2f, 2k, 2j, 2a, 2j, 2k, 2b, 2a, 2d, 2h, 2i, 2g, 2c, 2l, 2m, 3a, 3k, 3j, 3e, 3f, 3g, 3d, 3c, 3b, 3a, 3b, 3l, 3*, 3i, 3h, 3i, 3* 64 moves: 1b, 1h, 1i, 1g, 1j, 1g, 1i, 1d, 1c, 1l, 1k, 1a, 1b, 1*, 1e, 1f, 1m, 1d, 1m, 1k, 1n, 1a, 1n, 2f, 2e, 2f, 2b, 2k, 2j, 2a, 2j, 2k, 2b, 2d, 2a, 2d, 2h, 2g, 2i, 2g, 2l, 2c, 2l, 2m, 3k, 3j, 3e, 3f, 3d, 3g, 3d, 3c, 3l, 3b, 3a, 3k, 3a, 3b, 3l, 3*, 3i, 3h, 3i, 3* 41 moves: 1e, 1f, 1d, 1c, 1l, 1k, 1a, 1*, 1b, 1h, 1g, 1i, 1d, 1m, 1k, 1n, 2f, 2e, 2a, 2j, 2k, 2b, 2a, 2d, 2h, 2i, 2g, 2c, 2l, 2m, 3a, 3k, 3j, 3e, 3f, 3g, 3d, 3c, 3b, 3l, 3* 43 moves: 1b, 1h, 1g, 1i, 1m, 1d, 1c, 1l, 1k, 1a, 1b, 1*, 1e, 1f, 1d, 1m, 1k, 1n, 2f, 2e, 2a, 2j, 2k, 2b, 2a, 2d, 2h, 2i, 2g, 2c, 2l, 2m, 3a, 3k, 3j, 3e, 3f, 3g, 3d, 3c, 3b, 3l, 3*