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.
|Figure 1: Tiling a 6 x 10 rectangle|
|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|
 Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-02444-8
 Clarke, Andrew L. Polyominoes
 Muñiz, Alexandre. The Happiest and SaddestTilings, June 15, 2016