Polyhex Variegation


In his paper Polyominoes on a Multicolored Infinite Grid (in Thane Plambeck and Tomas Rokicki, eds., Barrycades and Septoku: Papers in Honor of Martin Gardner and Tom Rogers, Providence, 2020, MAA Press, Spectrum Series, v. 100, pp. 29–36), Hans Hung-Hsun Yu investigates how many colors are needed for the cells of the plane to ensure that a given polyomino has no two cells of the same color. Here I consider the corresponding problem for polyhexes.

A polyhex with k cells requires at least k colors. In the diagrams below, color counts that exactly meet this requirement appear in red.

For polyiamonds, see Polyiamond Variegation.


The dihex requires three colors:


For the A and I trihexes the same pattern suffices:

The third trihex requires four colors:


The same pattern admits three of the seven tetrahexes:

The I tetrahex requires five colors:

The Q and U tetrahexes require seven colors:

The J tetrahex requires nine colors:

General Result

A straight polyhex with an odd number of cells requires only as many colors as it has cells. The diagram below illustrates the typical pattern.

Last revised 2018-07-19.

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Col. George Sicherman [ HOME | MAIL ]