The Oakland Math Circle does Spot It

Rules of the game:

Each player gets one card. The deck goes in the middle. When you see a match between your card and the top card on the deck, you call it and collect that card, putting it on top of your pile.  Your card will always have a match with the center card, so it's just a question of who can spot their match first.

(There are 3 or 4 other games, but they're all pretty similar.)



After spending lots of time analyzing Spot It over the winter holidays, I thought it might make a good topic for the Oakland Math Circle. Two weeks ago, when I found out just a few hours ahead that I was scheduled to lead a circle, I jumped on BART, got off in downtown Berkeley, ran across the street to Games of Berkeley, bought 3 more tins of Spot It, ran back to BART, realized I'd left the tins on the counter at the store, ran back and forth another time, and made it to the circle just in time.

I had the students play a few rounds, and then we explored. The kids counted and found that most pictures appeared on 8 cards, but many appeared on only 7 cards. We eventually used Michelle's technique to figure out there were at least 57 pictures. (Pick a particular picture, the heart, for example. Pull out all the cards with a heart - there are 8 - and think about what we now know: 7 other pictures/card*8 cards + the heart = 57 pictures).

After a while, we focused on the question:  

How did the makers of this game make sure that every pair of cards has exactly one match?  

We didn't get much farther the first week. During the second week we had mostly new kids, so we started in the same place. But we got a bit farther, and tried to make our own cards with 4 pictures per card. We also had one group with a kid and an adult who had both worked on the problem before. They found a way to make every card in their deck match every other card. But their deck only had 5 cards. They figured out that they could make a similar deck of 9 cards with 8 pictures each. It wouldn't make a very satisfying Spot It game, though.

Yesterday was my third week of Spot It analysis with the Oakland Math Circle. We got back most of the kids from the first week, and played a few rounds again to get warmed up. (My first week with them, they seemed uninterested in math circle, this time they were really engaged, and much more fun to work with.) Then I let them each pick their color and gave them stacks of half-size index cards (3"x2.5"). They could choose to create cards using numbers or pictures, and were trying to make decks with 4 pictures per card, with one match between each pair of cards.

Lots of folks got the 5 card deck. We started calling that the minimal solution. I realized that was an easier solution to find than the solution that makes a bigger deck. (Although Chris and I never stumbled on it while we were creating our decks over the holidays.) One person pointed out that if they all matched on the same picture, you could have as many cards as you wanted. We called that the infinite solution. Since it would make a super-boring game, we added the condition that you have to use more than one picture for your matches, overall. People were so stuck on the minimal solution, I suggested starting with the infinite solution, and making a bunch of cards, trying to figure out when that would get you in trouble.

It turns out there are 13 cards in what I'll call the maximal solution. I realized that this brought up another question: Are there any symmetrical solutions with more than 5 cards, but less than 13?

One of my questions is whether the cards could use any number of pictures, or if there might be a constraint. (I'm thinking that the number of pictures per card might need to be 1 more than a prime, but I'm not at all sure. Yet.) Another cool aspect of this problem is that it illustrates the mathematical concept of duality. (I can't quite explain that yet, beyond saying there are 8 pictures per card, and we could have 8 of each picture in the deck.)

As I got the students into small groups yesterday, I told them, "I don't do math circles, I do math clusters." I find it's much easier to get lots of participation from the students if they're in small groups. Then my job is cross-fertilization. These last two weeks have been my favorite math circles yet. I think I'm finally getting the hang of it. My eternal thanks to Bob and Ellen Kaplan for helping me get started. I highly recommend their Summer Math Circle Institute, on July 8 to 14, in South Bend, Indiana. [Rodi Steinig, who went last summer, is doing great work in Pennsylvania, and blogs about her circles here.]

Each math circle leader has their own style, so if you're thinking about leading math circles, you need to find cool problems that work for you. My own favorites are: this problem (!), the magic pancake, playing with base 3 and base 8, and Pythagorean triples. You can find lots of math-tested-problems at the National Association of Math Circles site.