# Abe, Butch, Cam, and Dan

There are four men: Abe, Butch, Cam, and Dan. There are four hats: two white and two black. They each get a hat: Abe gets black, Butch gets white, Cam gets black, and Dan gets white. They don’t know what hat color they have. Abe goes into a room so that no one can see his hat. The others line up in front of the door in the order Butch, Cam, and Dan (Butch can’t see the others hats, Cam can see Butch’s, and Dan can see Cam and Butch’s hats). Whoever can figure out their hat color first wins \$1,000. If they are wrong they are out of the running.

Which man can figure out his hat color without a doubt?

### 4 guesses to Abe, Butch, Cam, and Dan

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• Tyler

Abe

• You’d think it would be Dan because he can see TWO hats in front of him, but because those two hats are different colors he can’t be sure which one he has. So as soon as Cam realizes that Dan does not know for sure, then he can deduce that he’s wearing the opposite color hat of the one he can see. It’s Cam.

• brent

It is not clear if the men know that there are exactly two black and two white hats? But I don’t think this matters.

As soon as possible any one (all) of the men should arbitrarily pick a hat colour and say that is what they have.
If they are 1st to register, they have a 50% chance at winning the \$1000. If they are wrong, they are simply eliminated from the running.

There does not seem to be any down-side to doing this, and the 50% up-side of winning \$500.

Given that the above behaviour makes sense to everyone (including Dan) – Cam does not have an advantage because Dan will answer right away so Cam does not know if Dan is just guessing, or really knows.

So that’s the best behaviour for the men. To answer the riddle of which man can figure out his hat colour without a doubt. The answer is all of them. If they do the above, and get the \$1000 they guessed correct and know their hat colour. If they are told they are incorrect and eliminated, they have still figured out their hat colour without a doubt. Just not in time to win the \$1000.

• Taylor, you are today’s winner!
Cam can figure it out. If Dan saw that Cam and Butch had the same color hat he would know his hat color. Since he doesn’t quickly figure out his hat color, Cam knows that him and Butch don’t have the same hat color. So all he has to do is say that his hat color is the opposite of Butch’s hat color (after giving enough time to be sure that Dan doesn’t know his own color).