## Dodecahedronquintomino Puzzle

John Horton Conway defines a "quintomino" as a regular pentagon whose edges (or triangular segments) are colored with five different colors, one color to an edge. Not counting rotations and reflections as being different, there are 12 distinct quintominoes. Letting 1, 2, 3, 4, 5 represent the five colors, the 12 quintominoes can be symbolized as follows:

A. |
12345 |
G. |
13245 |

B. |
12354 |
H. |
13254 |

C. |
12435 |
J |
13425 |

D. |
12453 |
K. |
13524 |

E. |
12534 |
L. |
14235 |

F. |
12543 |
M. |
14325 |

The numbers indicate the cyclic order of colors going either clockwise or counterclockwise around the pentagon [see Figure 12, left]. In 1958 Conway asked himself if it was possible to color the edges of a regular dodecahedron [Figure 12, middle] in such a way that each of the 12 quintominoes would appear on one of the solid's 12 pentagonal faces. He found that it was indeed possible. Can readers find a way to do it?

Figure 12

### The A quintomino The dodecahedron

Those who like to make mechanical puzzles can construct a cardboard model of a dodecahedron with small magnets glued to the inside of each face. The quintominoes can be cut from metal and colored on both sides (identical colors opposite each other) so that any piece can be "reflected" by turning it over. The magnets, of course, serve to hold the quintominoes on the faces of the solid while one works on the puzzle. The problem is to place the 12 pieces in such a way that the colors match across every edge.

Without such a model, the Schlegel diagram of a dodecahedron [Figure 12, right] can be used. This is simply the distorted

skeleton of the solid, with its back face stretched to become the figure's outside border. The edges are to be labeled (or colored) so that each pentagon (including the one delineated by the pentagonal perimeter) is a different quintomino.

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