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?

A Gaggle of Puzzles

What is the collective noun for puzzles, anyway? If there isn’t one, there certainly ought to be.

Here are a few puzzles with simple and thematically similar solutions. They require no math and are accessible to layfolk. I included sources where I could.



Save the World

Aliens have invaded and chosen you and 9 other people to determine the fate of the world. You can strategize beforehand.

• Each person is given a hat, red or blue.

• Person n can see the colors of hats 1, 2, 3, … n - 1 (but not their own). So, person 1 sees nothing!

• In reverse order, beginning with person 10, each person guesses the color of their hat. Everyone can hear what each person guesses.

• People can only say “red” or “blue.” They cannot communicate to each other using pronunciation, speed, etc.

• At the end, for each wrong guess, the aliens kill 10% of Earth’s population.

What is the maximum percentage of humanity that can be guaranteed to be saved?




Mutilated Chessboard (source, with solution)

A standard 8 by 8 chessboard has the lower left corner and the upper right removed, with the remaining 62 squares left intact. You are given 31 dominoes, each of which can cover 2 squares.

• The dominoes may be oriented vertically or horizontally.

• The dominoes may not overlap or go off the edge of the board.

Can you cover the chessboard with the dominoes?



Purple and Orange Polyhedra (source: 2013 HCSSiM Interesting Test)

I have a bin of containing purple and orange plastic polyhedra, 2013 of each. I have plenty of additional polyhedra of both colors, so I won’t run out. Every 17 seconds, I remove two polyhedra at random from the bin.

• If they are the same color, I put a purple polyhedron into the bin.

• If they are different colors, I put an orange polyhedron into the bin.

When there is only one polyhedron left in the bin, what is the probability that it is orange?



Chocolate Bar

You have an n by m rectangular Hershey’s bar, and you want to break it up according to the grid (i.e., in the usual way) to get n*m pieces.

• You can only break one piece at a time; you may not do things like stacking multiple pieces on top of each other and then breaking them.

What are the minimum and maximum numbers of breaks you can use to break up the chocolate bar?