## Archive for January 6, 2013

### Swish, Swash, I Was Doin’ Some Math

I’m not sure *what* I want to be when I grow up, but I know *who* I want to be. That would be Bill Ritchie, owner and founder of Thinkfun^{®}, a company that makes and markets games and puzzles, including the wildly popular Rush Hour^{®} game.

Our house has recently become addicted to Swish^{TM}, one of Thinkfun’s newest games. *Swish* is a spatial card game played with transparent cards that can be rotated and flipped to make swishes. Swishes — named after the sound made by a perfect basketball shot going through the net — are made by layering as few as two or as many as 12 cards so that every ball swishes into a hoop of the same color.

For instance, the two cards below can be combined to make a swish. The orange dot on the left card aligns with the orange hoop on the right, and the blue hoop on the left aligns with the blue dot on the right, which forms a double swish when the cards are placed one on top of the other.

Here’s a more advanced example. These four cards can be layered to make a quadruple swish:

The examples above show the cards in perfect alignment. They just have to be placed one on top of another for the dots and hoops to align. What makes the game fun and challenging is that cards typically aren’t in perfect alignment like this. One or more of the cards will have to be flipped or rotated to make a swish. In addition, the set-up arranges 16 cards in a 4 × 4 grid, so the two complementary cards are rarely next to one another. Consequently, the game uses all three geometric transformations — reflections, rotations, and translations.

Each card contains one hoop and one dot. These two objects are placed in one of 12 regions — each card is divided into an imaginary 4 × 3 grid. Theoretically, it’s possible to make a 12-tuple swish… though I’ve yet to pull off such a feat during a game.

There are a number of interesting math questions that can be asked about the game:

- Will there always be a swish in any 4 × 4 array of cards?
- The deck contains 60 cards. Does that account for every possible combination of hoops and dots? [The answer is obviously no, but that leads to some interesting follow-up questions.] How many different cards are possible? Finding the total isn’t as trivial as it sounds, since cards can be rotated or flipped; duplicates need to be removed. Which cards were included in the deck, which ones were excluded, and why?
- How many different double swishes can be formed by the cards in the deck? How many triple swishes? … How many 12-tuple swishes?

I don’t have answers to all of those questions yet. But I look forward to discussing them (and others) when I introduce *Swish* to some colleagues at an upcoming math conference!