### Finding 3-color mapped solutions to the 6x10 pentomino puzzle

There are exactly 12 planar shapes that can be formed by connecting 5 squares edge to edge with no overlap. These shapes were made famous by mathematician Solomon Golomb in 1953 who coined the term pentomino, provided mathematical and visual descriptions of the 12 pieces, and offered some games and puzzles using the set. A common puzzle challenge is to use all 12 pentominoes to tile a rectangle with no gaps or overlap. The set of 12 pentominoes contains 12 x 5 = 60 squares, and indeed one popular puzzle is to fit them into a 6x10 unit rectangle (Fig. 1). In 1960 mathematician Colin Brian and physicist Jenifer Haselgrove found that there are exactly 2339 possible pentomino arrangements that could fill a 6 x 10 rectangle.[1]

Figure 1: Tiling a 6 x 10 rectangle |

**such that similar colored pieces do not touch. Patrick Hamlyn showed that of the 2339 solutions only 94 could be colored in this way.[2] Further analysis performed in 2014 by Alexandre Muñiz breaks down the possible 3 color solutions and finds that only 53 of these 94 solutions color the pentominoes to “balance”, where balance indicates that the colors are distributed evenly to three groups of four pentominoes.[3] In preparing an educational video on pentominoes, it struck me that the 3 color solution would be the nicest to show, so I purchased a set of pentominoes produced by Discovery Toys that conveniently came in three bright colors (Fig. 2).**

*only three colors*Figure 2: Three color set by Discovery Toys |

After many unsuccessful trials I decide to find out
analytically if a 3 color solution was possible for this set. I found a list of
the 2339 solutions online and wrote some Visual Basic code that was able to
confirm the work of Muñiz finding just 53 balanced 3-color solutions out of
13,507,725 possible such combinations for the 6x10 tiling. See this appendix for
images of these 53 solutions.

However I also found that for each of the 53 balanced 3-color solutions there was exactly only one way to compose the 3-color sets of 4 pentominoes each. An example of a balanced 3-color solution is shown in Figure 3.

Figure 3: Example of balanced 3-color solution |

Figure 4: one and two solution puzzles (laser cut acrylic and wood)available here: 3-Color Pentominoes |

### Tiling a rectangle with balanced 6x10 3-color solutions

These 48 solutions can produce an aesthetic tilling of an 8
x 6 rectangular configuration of 6 x 10 solutions, making a rectangle 60 unit
squares wide by 48 unit squares tall. Coloring is still the only method used to
allow the recognition of the pentomino shapes, so we still require that each
pentomino not touch the side of a similarly colored neighbor, but we relax the
coloring restriction to what I will call loose coloring, in which we allow
corners of similar colored tiles to touch. Using such loose coloring allows the
eye to discern the boundaries between the 6x10 balanced color solutions, and at
the same time one can still clearly demark the individual pentomino tiles. There
are more than N=48! ways to order these 48 tiles, so it seems likely that
solutions might exist for the above describes criteria. A search using a simple
backtracking algorithm coded in Visual Basic found the following solution after
checking 103 million configurations (Fig. 5).

Figure 5: Aesthetic tilling of an 8 x 6 rectangular configuration of 6 x 10 solutions |

[1] Golomb,
Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton
University Press. ISBN 978-0-691-02444-8

[2] Clarke,
Andrew L. *Polyominoes*

[3] Muñiz,
Alexandre*. The Happiest and SaddestTilings*, June 15, 2016