A Drizzle of Puzzles

Today's forecast will include a light rain of puzzles. We are expecting at least 1", but it's harder to determine the upper bound.



Purplified Chessboard (source, with solution)

At time t = 0, some of the squares on my 8-by-8 chessboard inexplicably caught a disease which turned them purple. This disease is contagious: if at time t, a square is adjacent (either vertically or horizontally) to at least two purple squares, then at time t+1, the square becomes overwhelmed by purple-germs and turns purple itself. Once a square has been purplified, it never recovers.

I can tell that my chessboard will eventually become entirely purple. What's the largest possible number of non-purple squares my chessboard can have at time t = 0?




Guessing Game (source)

Aliens have invaded the planet again and chosen you and two other people to determine its fate. Your task is as follows:

• Each of you is given a hat, red or blue, with a 1/2 chance of each color. You can see the colors of the other two people's hats, but not your own.

• Without communicating anything to each other, each of you privately writes down either "my hat is red," "my hat is blue," or "I will not guess."

• If at least one person guesses, and everyone who guesses is correct, the aliens will leave you in peace. Otherwise, they will annihilate humanity.

You can discuss your strategy with the other two people beforehand. If you choose an optimal strategy, what's the probability that the world is saved?

Cutting a Cross (source, with lower bound solution)

You may make two straight-line cuts into the following cross. What is the largest number of pieces you can get?