Kadon Enterprises, Sextillions solutions by Jack Wetterer

Special solutions: Sextillions

The solutions below were sent to us by Jack Wetterer, associate professor of physics at the US Air Force Academy and a big fan of polyomino and polycube sets. Professor Wetterer liked the way we package Heptominoes as a three-part rectangle and worked out a couple of three-rectangle divisions for Sextillions. Notice that the middle ones have a hole that's the shape of one of the pieces.

Jack also liked our star-shape design for Poly-5 and built an even larger one with the combined sets, with co-solver Chris Patterson. Enjoy!

Three 6x12 rectangles, with the 11 unequal pieces all together in the center enclosing a hole like one of their own.

Three 9x8 rectangles, with a symmetrical center hole.

A PolyStar with the combined Poly-5 and Sextillions set, going outward from smallest to largest.

Later we received this awesomely symmetrical solution from Darian Jenkins. He tells us that he used Gerard's Universal Polyomino Solver to find all 4579 solutions to the pentomino-tetromino part. The hexomino part has over 20,000 solutions!
A PolyStar with the combined Poly-5 and Sextillions set, with concentric symmetry.

A PolyStar with the combined Poly-5 and Sextillions set, with concentric symmetry.

Another time, Jack Wetterer sent us these two rectangular solutions that allow division of the Sextillions set into 3 subrectangles, with the 12 "unequal" pieces all contained in one group. Jack custom-ordered these in the color divisions you see here, to match the style of his Heptominoes and Octominoes.

Sextillions solution by Jack Wetterer, showing 3 subrectangles, 8x9, by color

In 2021 we got the biggest surprise after 37 years of making the Sextillions set. Another dedicated solver, Doug Caine, did an in-depth research into the possibility of improving our original packing pattern with the "unequal" group completely separated and not even touching the center window. His computer program found that such a solution does exist, unique except for switching a couple of pieces. Here they are. Can you spot which pieces traded places? Hurray, Doug!

Sextillions solutions by Doug Caine, with full separation of unequal group Second solution by Doug Caine, 2 pairs exchanged

Another great discovery in 2021 with the hexomino set we call Sextillions was made by Livio Zucca, a world-class polyomino researcher. The 35 hexominoes have a little oddity about them, containing 11 shapes that, if checkerboarded, yield 4 and 2 squares each. The other 24 divide evenly into 3 and 3. We color them accordingly.

This "parity" glitch makes some solutions impossible, especially rectangles, and the 11 "unequal" pieces usually need to be used in pairs for balance. But that 11th piece is a stumbling block. Our way of coping with this lonely troublemaker was to add one duplicate piece, the mirror image of one of the 11.

Livio's tour de force was to find, not just one, but two, themes in which all 35 shapes are consistent. The first one frames a three-fold outline of each shape in a well-proportioned frame, with the piece of that shape parked outside. The second builds a five-fold model of each shape, plus a neat 9-piece pedestal to hold the single of that shape. All 35 pedestals are congruent, whether the piece being modeled is equal or unequal!

Sextillions solutions by Livio Zucca, 35 framed hexomino spaces with their own shape as singles

Sextillions solutions by Livio Zucca, 35 order-5 hexominoes plus a 9-piece pedestal holding its single

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