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Rook and pebbles

I have a rook on an infinite chessboard. On each of my turns, I can move it as far as I want in any of the cardinal directions; on each of your turns, you can place n pebbles that block my movement. For what values of n can you trap me?

(Obviously, 4 will do. Almost-equally-obviously, 3 will do. But 2? 1?)

The circular lake

Rosa and Sam are visiting a perfectly circular lake. Sam is swimming; Rosa is on the shore. Sam would like to get out, but he'd prefer Rosa not be there to greet him when he reaches shore, because she's carrying a chainsaw. Sam can swim freely in the lake, which Rosa can't enter, because she's carrying a chainsaw. If Sam can reach the shore, he can run faster than Rosa, because she's carrying a chainsaw. But, even chainsaw-laden, Rosa can run \(v\) times faster than Sam can swim. For what values of \(v\) can Sam escape?

EARTH / HEART

A monkey is hammering on a typewriter, producing a stream of independent uniformly-random draws from the letters A-Z.

  1. On average, how long will it take for the monkey to type the word EARTH?
  2. On average, how long will it take for the monkey to type the word HEART?
  3. If we stop the monkey as soon as it types either of those words, how likely is it to be HEART?

The pirate ship

There's a pirate ship at some integer position on the real number line, sailing along with a constant integer velocity. You don't know where it is; you don't know its velocity. At each of t=0, t=1, t=2, etc., you can fire a torpedo at one point on the number line. Come up with a strategy that is guaranteed to eventually hit the ship.

(Extra credit: the pirate ship has some finite-but-unknown size, but its velocity and starting position aren't guaranteed to be integers. Come up with a strategy that will eventually hit it.)

(Extra credit: as above, but in 3D space instead of 1D.)

Infinite chess

I think that, in chess, one says that White can "force mate in \(n\)" if White has some strategy that is guaranteed to checkmate Black in \(n\) or fewer moves.

Consider an infinite chessboard, with a possibly-infinite number of pieces (but only one king per side, to keep "checkmate" well-defined). Find a board-state where White can force mate (i.e. no matter what Black does, the game is guaranteed to terminate), but there's no \(n\) such that White can force mate in \(n\).

We might say, there, that White can "force mate in \(\omega\)," \(\omega\) being defined as the smallest thing that's greater than all natural numbers. Which raises the natural follow-up questions: is there a board state where White can force mate in \(\omega + 1\)? \(2 \omega\)? \(\omega^2\)? Any arbitrary ordinal number?