Dezign-8: a gallery of variations




These are just a a few of the many decorative patterns you can create with the Dezign-8 set of tiles. The more symmetrical you make them, the prettier they are, although totally random arrangements can also look striking. The only requirement is that no open path ends remain — that every path connects to another one. Notice how each design has as many "groups" as it has "loops." A group is a separate, connected path network. A loop is an enclosed area. Mathematicians explain this phenomenon in terms of graph theory, a fortuitous selection of the right combination of tiles. We don't quite understand the theory, but in practice it's one of nature's lovelier phenomena.

The Dezign-8 booklet reports that from 1 to 16 groups can be formed. On September 24, 2005, this claim was disproved when Daniel Austin of Maryland, then a mere 14 years old, in our Gamery pavilion at the Maryland Renaissance Festival found a solution with 17 groups and loops, illustrated below. This discovery created great excitement for setting a new record, and we heartily congratulated Daniel on this achievement.


A little over a week later, on October 2, 2005, Daniel emailed us to say he had surpassed his own record with an 18. Here's the signed and dated drawing he sent. All we can say is "Wow!"


To top it all off, on December 29, 2005, Daniel sent word that he and his brother Ben had found a 19, and provided an elegant proof that this was, indeed, the max. Thus encouraged, Kate set out to meet that challenge, and her result appears below right. It is not a unique solution. If you find a different one, we'd like to see it. Daniel's solution arrived shortly thereafter, shown below left.

19-group, 19-loop symmetrical solution by Kate

19-group, 19-loop symmetrical solution by Daniel

And now for the ultimate record to date, a solution by Daniel with 19 groups, 19 loops, and symmetry on both the horizontal and vertical axes:


Embedded pentominoes
In 2017, during his year as a Kadon consultant, puzzle solver extraordinaire Alex Streif dove into further possibilities by embedding our favorite shapes, the pentominoes, within Dezign-8 solutions. Here's that collection. The first nine are by Alex. Kate finished off the last three in September 2020 to submit the whole collection to the 2021 JMM art exhibition, with this little write-up:







Then Alex managed to pack more than one pentomino into the tray:


And for good measure, Alex came up with these pretty symmetries:


This last one is actually an embedded hexomino (6 squares joined). There are 35 hexominoes in all. Wonder how many of them can be embedded? Setting and breaking records is fun. There's no end to the phenomena to be explored. All we have to do is keep asking interesting questions and pushing the limits...

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