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Math Circle Problem: Analysis of the Game “Spot it!”

This problem was originally posed by Sue on her blog Math Mama Writes (and was presented by her at the Match Circle Institute). There’s a kids’ game called Spot It, where there are cards with pictures all over them in a pile. If you have a match with the pile in the middle, you call out the name of that icon and grab the card. The person who collects the most cards win.

But here’s the interesting part – despite there being 57 different pictures and 55 different cards, every card has one and only one match with every single other card.

How did they make this game?
Would it work for every number of pictures?
Is there an algorithm for every number of pictures?

The best way to see why this is such an interesting question is by trying to make your own deck with 3, 4, 5, 6 etc different pictures on each card. If you do that, my “solution” below might make some sense!

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