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12 Rovers

Two rovers will be launched to a perfectly spherical planet. When they land, they will have no way of determining where on the planet they are, nor what direction they face. Each has a radar that allows it to detect the other rover, but the radar has a range of only one mile. The diameter of the planet is known. Program the rovers so that they meet. The two rovers must run identical software.

11 One hundred and one dalmatians

One hundred and one dalmatians have the property that if you remove any one of them from the set, the remaining hundred can be divided into two groups of fifty each, so that the sums of the numbers of spots on the dogs in the two groups are equal. If every dalmatian has the same number of spots, this property clearly holds. Is it possible that some of the dalmatians have different numbers of spots?

10 Tying together spaghetti

A bowl has 100 strands of spaghetti in it. You reach into the bowl, randomly grab two loose ends, tie them together, and repeat the process until there are no loose ends left. How many loops do you expect to end up with?

9 Flipping coins around a table

You and nine other prisoners are seated around a table. Simultaneously, each of you flips a coin. Everyone can see the results of everyone's flips except his own. Simultaneously, you all guess how your coin flips turned out. You win collectively if you are all right, otherwise you lose collectively. You may collude beforehand.

8 Cover a chess board with dominoes

I have a chess board with one pair of opposite corners removed. You have dominoes, each of which is the size of two squares of the board. Can you cover the board using the dominoes?

7 Cups on a round table

You and I will play a game taking turns placing round cups on a round table. The first player that doesn't have room to place a cup loses. Would you rather go first or second?

6 Two envelopes

I have two sealed envelopes with money in them. One contains twice as much money as the other. You may pick one and look inside. You then decide whether you want to keep the money in the envelope you opened, or take the money in the other one.

5 \(n\) prisoners with colored hats

You and 99 friends will be given hats, each of which is one of 100 distinct colors. You cannot see your own hats, but you can see each others'. Simultaeously, you will all guess the colors of your own hats. You all win if at least one of you is right. You may collude before being given the hats, but cannot communicate afterward.

4 Two prisoners with colored hats

You and a friend will be given hats, each of which is either black or white. You cannot see your own hats, but you can see each others'. Simultaeously, you will both guess the colors of your own hats. You both win if at least one of you is right. You may collude before being given the hats, but cannot communicate afterward.

3 Guess the polynomial

I am thinking of a polynomial \(f\) with non-negative integer coefficients. You may give me an integer \(x\) and I will tell you \(f(x)\). Guess \(f\) in as few queries as possible.

2 Flipping coins blindfolded

In front of you are 9 heads-up and 6 tails-up coins. You are blindfolded. Divide the coins into two piles which have the same number of heads-up coins.

1 Bloodthirsty pirates' loot

100 pirates are deciding how to divvy up 100 coins. The highest-ranking pirate proposes a distribution scheme, and all of the pirates vote. If strictly more than half vote no, then the proposer is killed, and the process continues with the next-highest-ranking pirate, and so on. The pirates value survival, coins, and bloodshed, in that order. What happens?