Nine Cards

You and I are playing a game. It’s a simple one: Spread out on a table in front of us, face up, are nine index cards with the numbers 1 through 9 on them. We take turns picking up cards and putting them in our hands. There is no discarding. The game ends in one of two ways. If we run out of cards to pick up, the game is a draw. But if one player has a set of three cards in his or her hand that add up to exactly 15 before we run out of cards, that player wins. (For example, if you had 2, 4, 6 and 7, you would win with the 2, 6 and 7. However, if you had 1, 2, 3, 7 and 8, you haven’t won because no set of three cards adds up to 15.) Let’s say you go first. With perfect play, who wins and why? [Solution]

Century Product

Given any three random integers — X, Y and Z — what are the chances that their product is divisible by 100? [Solution]

Island Trees

There is a single “tree” of $N$ islands, bridged together so that there is exactly one non-doubling-back path between any two islands. Each island has a $p$ chance of being destroyed today, taking its bridges with it. How many separate island-trees are expected to result? [Solution]

Street Grid

You’ve just been hired to work in a juicy middle-management role at Riddler HQ — welcome aboard! We relocated you to a tastefully appointed apartment in Riddler City, five blocks west and 10 blocks south of the office. (The streets of Riddler City, of course, are laid out in a perfect grid.) You walk to work each morning and back home each evening. Restless and inquisitive mathematician that you are, you prefer to walk a different path along the streets each time. How long can you stay in that apartment before you are forced to walk the same path twice? (Assume you don’t take paths that are longer than required, and assume beaucoup bonus points for not using your computer.) Extra credit: What if you instead took a bigger but more distant apartment, M blocks west and N blocks south of the office? [Solution]