And here are two solutions by Dan Austin with the rarest of all, six-fold symmetries, both rotational and mirror.

How many cubes?
This solution full of optical illusions is by Nancy Koles. We count a total of 25 cubes, many of them overlapping. Can you spot them all?

Opposite-color symmetry
Sometimes called "anti-symmetric" patterns, such arrangements are strongly reminiscent of the yin-yang concept:

Hexagon patches
Here are a couple of strongly symmetric designs by Eric the Wise, with hexagon regions beautifully balanced:

Intarsia's simplicity—just two kinds of tiles—illustrates dramatically the amazing capacity of life, the Universe, and everything to build to complexity from varying combinations of the most elementary particles. At the human level of just 32 pieces that you can manipulate with two hands and move through space on the table to reorganize at will, you can proceed at your own comfortable pace.

To calculate the total number of possible placements in the tray takes heroic mathematical calculations. We thank our friend, Eric Postpischil, for contributing some astronomical combinatorial statistics. And Henrik Morast worked out the analysis that yielded a mind-boggling total of 10 to the 24th power. That's a 1 followed by 24 zeroes!

Just see how many ways one pair of tiles can join to form a single hexagon. If we call the two tiles A and B, then we can join AA, AB and BB. In each pair a tile can be either side up, giving us 4 different combinations for each of the three pairings, or 12. But a hexagon can have 6 orientations, so we get 12 times 6, or 72, including rotational equivalents. And we're only at the first step.

How many ways can the 32 tiles be paired? That's 32-factorial, or 32x31x30x29... It's nice to contemplate that we have so many options, so many choices with which to bring our vision into reality.

Gene Roddenberry, the creator of Star Trek, had a great motto: IDIC — "Infinite diversity from infinite combinations." We rediscover the truth of that with every Intarsia arrangement. Even when the total of possibilities is beyond the range of our ability to encompass, we can glimpse it and play with it in beautiful ways. Our minds can touch the infinite and feel its echo in ourselves.

Happy puzzling!

Intarsia is available by custom order as a framed, printed 18x24" poster for $65. If you are interested in acquiring the poster, email us:
Intarsia print.

Here's a touching story. The solution at right was found by Robin King, wife of Irishman John McCallion. They met when John visited New York after some years of playing postal Chess across the Atlantic. Both were world-class game enthusiasts, and inevitably they married in 1990 and John moved to the U.S. Soon the two became game reviewers for Games Magazine, and continued as fans of Kadon games and puzzles.

Tragedy struck 23 years later, when Robin needed a lung transplant and did not survive the surgery in 2013. John was devastated and gave up his game collection and moved back to Ireland where his large clan provided some comfort. He did take up jigsaw puzzles, the more complex the better. He complained only when there were images with large areas of the same color, as those give no hints of how pieces joined. Solving those was a pain. As a special gift, Kate commissioned a custom jigsaw puzzle of Robin's Intarsia solution, with large areas of a single color. After tremendous effort, John conquered it and had it preserved by his niece for permanent display. Here he is, holding his treasure in memory of his beloved Robin.