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## Snake Puzzles and Solutions

A popular puzzle I have is called a "Snake Puzzle". It is composed of a string of 27 cubes connected by a shock cord; in each cube the cord either passes straight through or turns a corner, and the cubes can swivel relative to each other. The trick is to assemble it into a cube, like so:

There are several different makes of this puzzle, it also goes under the name Cubra (c).

If you are interested in graph theory these puzzles are interesting as they form Hamilton paths of the 3x3x3 grid graph. One can study which snakes have solutions and which do not, for example one can prove that it is impossible to solve the 3x3x3 cube with a snake composed of all corner cubes. See [1] for more on some of the math behind these puzzles. I generated the solutions to the puzzles shown here by a backtracking program.

A solution can be described by a string of "directions" that the cube follows when you wrap it into a cube, either Right,Left,Up, Down, Forward, or Back. So for example if a solution starts R R F L ... then put the end of the snake with the rest trailing off to the Right, then the next cube goes in the same direction (it must be a straight-through cube), the next cube (a corner cube) turns to point Forward, then it goes Left, etc.

The five Cubra puzzles I have seen are as follows, with several solutions for each:

Solutions:
R R F L U U F D D R U B B L L F D F U U B B R R F F
R R U L F F U B B R F D D L L U B U F F D D R R U U

Solutions:
R R F F L U U B D D L F U B B R R F F U B B L L F F
R R U U L F F D B B L U F D D R R U U F D D L L U U

Solutions:
R R F L U U F R D D L U L D B U B R R F U B L L F F
R R U L F F U R B B L F L B D F D R R U F D L L U U

Solutions:
R F L F R R U U B L F L D R B R D B U U L L F D B R
R F L F R R U U B L F D L U B D B U R R D D F U L B
R F L F U U R R B L D F D R U B D B U U L L F D B R
R F L F U U R R B L D L U B D R U R D D F F L U R B
R F R B U U L L F R F L D R B R U F D D L L B U B R
R F R B U U L L F R D F D R U B U F L L D D B U B R
R U L U F F R R D L B L F D B R F R B B U U L F R D
R U B D L L F F U B R U R B L D L U F F R R D L D R
R U L U R R F F D L U B L F D B D F R R B B U F L D
R U L U R R F F D L U L B R D R B D F F L L U B D R
R F D B L L U U F D F U R D B D L F R R U U B L B R
R U R D F F L L U R U L B R D R F U B B L L D F D R
R F U L D D B B U R D R U F D L F R U U B B L F L B

Solutions:
R R F L F R U L U R B B D F L U B L F F D D B U B R
R R F U B L F U L F D D R B L U B U R R F F L D R D
R R F U B L F D L F U U R B L D B U R R F F D L D R
R R F U B L U R F L D D L F U R D R U U L L B D B U
R R F U B U F L F R D D L U B D L F U U B B R D L F
R R F U B U F L B D F F L D B R F R U U L L B D B U
R R F U B U L D L U F F R D B U R F D D L L U B D R
R R F U B U L D L U F F D B R D L F R R U U B L F D
R R F U B U L D L U F F D R U B R F D D L L B R U L
R R F U B U L D L U F F D R U B R F D D L L B U R D
R R U F D F L B L F U U B R F D R U B B L L D F R B
R R U F D F L B L F U U B R F D R U B B L L D R F L
R R U F D F L B L F U U B D R B L U R R F F D L U B
R R U F D F L B L F U U R B D F R U B B L L F D B R
R R U F D F U L U R B B L F D B L U F F D D R B L U
R R U F D F U L D B U U L B D R U R F F L L D B D F
R R U F D L F R U L B B L U F R B R F F L L D B D F
R R U F D L U F L U B B R D L F D F R R U U L B R B
R R U F D L U B L U F F R D L B D F R R U U B L B R
R R U L U R F L F R D D B U L F D L U U B B D F D R

The wooden version at the top of the page looks like this when straightened out:

This is identical to the blue Cubra snake above and thus has the same solutions.

Jaap's Puzzle Page has a far more thorough analysis of snake cube puzzles, including a nice Javascript snake cube.
[1] F. Ruskey, and J. Sawada, Bent Hamilton Cycles in Grid Graphs, Electronic Journal of Combinatorics, #R1, 10 (2003) 18 pages.

2003, M. Weston