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PATH PUZZLES

Harry Nelsen, a former editor of the Journal of Recreational Mathematics (now edited by Charles Ashbacher and Colin R J Singleton), visited me last week.  He showed me several of the puzzles he'd helped create for Binary Arts, including Stormy Seas.  I'd seen the puzzle in stores, but didn't fully appreciate it until I'd tried it out.

One of the puzzles he showed me were Line Puzzles (Buried Treasure, Space Junk, Frog Feast) developed by Bill Darrah.  Each consists of eight domino-like cards, with paths on them.  The goal is to make a single path with these eight cards.  I was surprised at how difficult these puzzles are.  So, I decided to analyze them this this week.  There are nine dominoes with a single path, as seen above.  I thought about asking for a closed path with all nine dominoes, but found it was impossible.  The reason: the nine dominoes make a path of length 15, and no closed loop can have that length due to parity.  Juraj Lorinc, Roger Phillips, Federico Ricci-Tersenghi, Joseph DeVincentis, Andy Liu, Carlos Penedo, Jukka-Pekka Ikaheimonen, and Dave Tuller solved the problem.

There are 27 different ways to put 2 paths on a domino.  You can see them above.  What closed loop can be made with the full set of 36 path dominoes (1 & 2) that goes through the most dominoes (the most double visits)?  Roger Phillips solved many path problems.  The solutions are below.

Sometime next year, these pieces will be released on the market, my sources tell me.  There are 11 ways to put three paths on a domino.  Going to a 4x4 square, there are 35 different ways to put four paths on a square.  I published this as a game called Psliders a number of years ago.  It was independently discovered and published as the game Knots, with better rules but an incomplete set.

Roger Phillips has found a way to put all of the possible domino paths into a 10x10 square with a single path.

A tiling puzzle by Serhiy Grabarchuk is here.  Can anyone improve on his answer?

Kaliko (by the excellent Kadon Enterprises) and Tantrix are two different path games involving hexagons.  Mike McManaway, who invented Tantrix, sent me two images for the longest known simultaneous paths for the 56 Tantrix tiles.  The Tantrix site has very nice online puzzles and games.  A different path game is available from CheapAss Games - The Very Clever Pipe Game.
 The 4 LONGEST LOOPS * red = 33  * yellow = 30  * blue = 35  * green = 38  * total length = 136  found by Milan Kuchtiak * Spisska Nova Ves * Slovakia * September 1998 The 4 LONGEST LINES * red = 34  * yellow = 37  * blue = 35  * green = 40  * total length = 146  found by Jamie Sneddon and Paul Martinsen  * Auckland * New Zealand * April 1998