Hector's Puzzled

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

What are these bits? [Solution]

You’re going to play a game. Like many probability games, this one involves an infinite supply of ping-pong balls. No, this game is not quite beer pong. The balls are numbered 1 through N. There is also a group of N cups, labeled 1 through N, each of which can hold an unlimited number of ping-pong balls. The game is played in rounds. A round is composed of two phases: throwing and pruning. During the throwing phase, the player takes balls randomly, one at a time, from the infinite supply and tosses them at the cups. The throwing phase is over when every cup contains at least one ping-pong ball. Next comes the pruning phase. During this phase the player goes through all the balls in each cup and removes any ball whose number does not match the containing cup. Every ball drawn has a uniformly random number, every ball lands in a uniformly random cup, and every throw lands in some cup. The game is over when, after a round is completed, there are no empty cups. How many rounds would you expect to need to play to finish this game? How many balls would you expect to need to draw and throw to finish this game? [Solution]

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]

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]