One of my favorite games is Set. When I first discovered it, in 1998, I made a Java applet since I couldn’t immediately find the cards. It’s apparently been used as an example in a college course, which is pretty surprising given that it’s undergraduate Java (to be fair, I had hacked on it occasionally since). My applet used to be a fairly accurate representation of the original game, but a few months ago I got a letter from the lawyers for the company who make Set. They were pretty polite, but I had to change the graphics. So now my applet has polka-dots.

Set is played on ℤ_{3}^{4}. There’s a variant
called Projective
Set, which is
played on something called a finite vector space —
𝔽_{2}^{6}. Basically, that’s the six-bit
numbers (but Projective Set excludes the zero card). The rule is that
N cards can be removed if, for each bit position, the cards sum to zero
mod 2. Or, perhaps more simply, each symbol appears an even number of
times. Any seven distinct cards contain at least one such group.

These four cards can be removed.

One of Danielle’s co-workers lamented not being able to buy a copy, so I decided to make some. Unlike the one shown on the Wikipedia page, mine are accessible to the colorblind. I guess I should note that my version is in fact just called “Projective”, as Set is a trademark of Set Enterprises and I don’t want to annoy their lawyers any more than my applet already has. So now you can buy a copy. I’m not making money off of this because doing so would be a hassle; the price is what The Game Crafter set. I got mine yesterday, and it looks pretty good. Danielle beat me up some, and I enjoyed it.

This got me to thinking about what other mathematical objects could be
used for pattern recognition games. I immediately thought of
quaternions, and then got
Hamilton stuck in my
head. Did I mention that I’m not great at math? I had forgotten that
the quaternions are non-commutative, making the game much trickier to
design. But I guess I don’t need to be totally accurate to the math.
As long as I keep i^{2}=k^{2}=k^{2}=ijk=-1,
everything will probably work out. The idea will be to find N cards
whose product is one. I’m thinking of calling it Uno.

I even came up with some icons, based on the Swedish point of interest symbol:

Notice that each of i, j, and k can be combined with a rotation to form the -1 symbol, and i, j, and k can be overlapped to do the same.

This might be too easy, so maybe I’ll need to do ℚ^{2}
or something. Or just go straight to 𝕆, whose multiplication
table is too big to remember. But you can always use this
simple and easy to understand diagram (from Wikipedia):