Warning: this essay contains solutions to math problems.
Here is a famous hat puzzle:
A king decides to give 100 of his wise men a test. If together they pass, they can go free. Otherwise, the king will execute all of them. The test goes as follows: the wise men stand in a line one after another, all facing in the same direction. The king puts either a black or a white hat on each wise man. The wise men can only see the colors of the hats in front of them. In any order they want, each one guesses the color of the hat on his own head. Other than that, the wise men cannot speak. To pass, no more than one of them may guess incorrectly. Given that they have time to agree on a strategy beforehand, how can they assure that they will survive?
Instead of discussing the puzzle above, I’d like to look at a different version. It is an infinite variation of the puzzle that my son Sergei brought back from the Canada/USA Mathcamp last year.
The king has a countable number of wise men. The line starts from the left and is infinite in the right direction. The wise men are all facing to the right and they see the infinite tail of the line. Again, the king places either a black or white hat on each head and they can only say one of two words: black or white. Will they be able to devise a strategy beforehand that ensures that not more than one person makes a mistake?
Oh, I forgot to mention: you are allowed to use the axiom of choice.
Here is the solution. You can build an equivalence relation on the possible placements of hats. To be equivalent, two ways of placing the hats should have the same tail. In other words, there is a person such that both hat arrangements to his right are the same. By the axiom of choice you can pick a representative in any equivalence class. The first wise man looks at all the other hats and calculates in how many places the tail differs from the representative of the class they picked. This is a finite number, and by stating one color or the other, he signals the parity of that number. After that, all the wise men say their colors from left to right. Everyone sees the tail and everyone hears the color choices of the people behind. So every wise man can reconstruct the color of his hat with this information. Only the first person may potentially be mistaken.
Many things about this solution bother me. Where is this country that can fit an infinite number of people? What kind of humans can see into infinity? How much time will this procedure take?
Aside from the practical matters, there are mathematical matters that bother me, too. By the axiom of choice you can pick an element in every class. The problem is that all of the wise men have to pick the same element. The axiom of choice claims the existence of a choice function, which picks an element in each set. So the function exists, but can we distribute this function to many wise men? Remember, they need to agree on this function the night before.
We already implicitly assumed that our wise men have a lot of magical abilities. So we can add to those the ability to go through all the possible tails and memorize the representatives for all the tails in one evening.
But still, I am very curious to know what follows from the axiom of choice. Tell me what you think: does the axiom of choice imply that we can distribute the choice function, or do we need a new axiom? In your opinion, will these wise men live?Share: