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Last month, I was introduced to the card game SET. The game raises several interesting questions eg what's the probability that n randomly drawn cards contain k sets, is it possible to end with 12 (or 15) cards which contain no sets... I would like to present the solution to a question like this to undergrads, and through it introduce them to an area of mathematics. For example, as mentioned previously, there is the issue of maximal caps, which Davis and MacLagan's paper relates to projective space and error-correcting codes. Can you give me any similar ideas? Perhaps relating to a variation of SET listed here?

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    @Willie: I hate to repeat myself. I only said that the question in the comments should be opened in a separate post. Then it should be made CW. I didn't tell her to do it by herself.2010-12-27

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While I personally don't remember the mathematics, I can give you two leads which, if you follow it up, should give you some ideas.

  • Try contacting Jordan Ellenberg. Several years ago he gave a talk to the Princeton University Undergraduate Math Club on precisely what you described in your question. He will definitely be able to give you some pointers.
  • Also, try asking Aaron Potechin. he is an expert Set player and I remember seeing a pre-print of his on the Maximal Cap issue a few years ago, when he was still an undergrad.
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    BTW, Jordan's talk introduced, if I remember correctly, vector spaces over finite fields, and from there led to a nice discussion of arithmetic geometry. The ideas are very accessible, but has enough non-intuitive "Whoa" moments that made the talk very engaging.2010-12-27