I'm sorry, math-fun, I cheated on you. I recently played a cool card game called Spot It. It consists of a deck of 55 round cards; each card has eight assorted symbols on it. The deck has the property that every pair of cards shared exactly one common symbol, and they build a few games around the spot-the-shared-symbol mechanic. It's a lot of fun, and I've been playing it regularly with my 5-year-old and 12-year-old kids. The only annoying part? The any-two-cards-share-one-symbol thing is encoding point-line incidence in the projective plane over the field with 7 elements... in which there are fifty-SEVEN points. The manufacturers, for reasons I can't begin to guess, came up two cards short. (The cheating that I mentioned is because I wrote all this up, complete with pictures -- and posted it on Google Plus: https://plus.sandbox.google.com/114937925666302803969/posts/Fa9Znut5JmK. Go take a look -- I'm enjoying G+, and hey, my math-geeky post has been reshared 20 times now, with lots of non-math-focused people reading it. So there's something worthwhile going on in this slice of the social networking world.) So how does this generalize to n symbols per card? Any time n-1 is a prime power, the projective plane construction works, giving you n^2-n+1 cards (points) each showing n out of n^2-n+1 symbols (lines they're incident to). But when n-1 is not a prime power, e.g. when n=7, what can you get? That is, what's the largest finite geometry with a line between each pair of points, and at most n=7 lines incident to each point? Surely block design people must know the answer to this sort of question, but as far as I can tell, the OEIS doesn't have this. --Michael -- Forewarned is worth an octopus in the bush.