Kadon Enterprises, Inc. Quintillions, the symmetrical pairs

The symmetrical pairs

The 12 uniquely shaped quint pieces (pentominoes) can be paired in countless ways, but only the 52 pairs below are symmetrical with at least 2 unit edges of contact. The 52 pairs form 34 distinct shapes — some can be built with more than one pair of quints, and some pairs of quints can form two or even three different symmetrical shapes.

Just for fun: Can you draw the 10 symmetrical pairs that have only 1 unit edge of contact?

The 52 distinct symmetrical pairs of quints

Now we can ask if there is any way to arrange the 12 quints into 6 simultaneous pairs, and the answer is surprising. Yes, in 2153 different ways! In only one of those, every pair can make more than one shape. In two others, no pair is diagonal. There are eight divisions where every pair makes only one shape, and there is no way for all 6 pairs to be diagonal. We thank Jim Kolb and his BASIC program for these statistics, done way back in September 1981 and first published in Quint-Gram No. 4, Spring 1983.

It's fun to see what you can build with six pairs, as both two-dimensional and three-dimensional constructions. The Quintillions booklet illustrates several.

And here's how we package the Quintillions set. Can you spot all 5 symmetrical pairs, plus 3 symmetrical triplets? It was a happy coincidence that the W, our logo, ended up in the center.

Quintillions packing pattern with 5 symmetrical pairs

This entire fascinating avenue of investigation became immortalized in an article Kate contributed to the Australian Game & Puzzle Design journal of the Queens University of Technology, published in Volume I, No. 1, June 2015, pp. 44-49. Read "Symmetrical Pentomino Pairs" here, courtesy of Cameron Browne, Editor-in-Chief.

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