Heptominoes

Heptominoes

Polyominoes

The word polyomino was coined by Solomon Golomb in 1953. A few years later, Martin Gardner popularized polyominoes (in his December 1957 Scientific American column). The number of "free" (i.e., free to be flipped over) n-ominoes for n = 1 to 7 are: 1, 1, 2, 5, 12, 35, and 108. It's an easy exercise to show that these 164 n-ominoes have a total area of 1055. In an old issue of the Journal Of Recreational Mathematics (Volume 13, No. 1: 1980) I asked if the 164 pieces would fit into a 5 by 211 rectangle. One year later, the journal published a solution by Bernard Rosenheck.

In my picture of the 108 heptominoes, there is a monomino-hole in the center of each coloured rectangle. The one in the middle is inherent in the sole heptomino that has a hole, the one on the right is created by two adjacent heptominoes, and the one on the left is created by three adjacent heptominoes. The picture nicely illustrates that 108 ∗ 7 + 3 = 3 ∗ 11 ∗ 23.

Game and puzzle company Kadon Enterprises sells lasercut acrylic polyomino sets (Poly-5, Sextillions, Heptominoes, and Octominoes). Some gorgeous multi-set constructions can be found here. I especially like Karl Wilk's "pentomino clock".

Thursday, January 8, 2009