So here's something interesting. I played a game earlier today. Don't remember the name. The basic idea is that there's some cards, each with 8 objects on, and for each possible pair of cards, exactly one object is on both cards, which players are trying to find.
Here are the numbers the rulebook gives: 55 cards, 8 objects on each card, 50 objects total. So naturally, as a mathematician, I decided to try and calculate an upper bound on the possible number of cards (and from there intended to see if I could narrow it down to an actual maximum. And I got 43 as an upper bound, which... well, obviously slightly worried me.
My method was this: One card has eight objects. That means, for any possible pair of two objects on that card, that pair can't appear on another card (or it'd have two objects in common with the first card). As there are eight objects, that means there are (8C2) = 28 different pairs of objects covered, none of which are shared with another card. There are 50 objects total. That means there are (50C2) = 1225 total pairs of objects. So now as every card has 28 unique pairs, we can obtain an upper bound by taking (total no. of object pairs) / (pairs on each card) = 1225/28 = 43.75, and since that's an upper bound we can take the integer part of 43.
So... where did my reasoning go wrong? Or is it solid, and the information the game gave perhaps wrong? By my calculation, with at least 55 objects this method gives an upper bound of 55, so it's possible the game's statement of '50 objects' really meant 'over 50'.
