# The Center of Math Blog

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## Monday, February 23, 2015

### Problem of the Week

 Click to enlargen!
Happy Monday from the still-frozen tundra of New England. Like each Monday, I have shared the above problem to our Twitter, Google +, and Facebook pages. Today's problem, more logic than straight mathematics, was adapted from this puzzle site. I had actually encountered this problem once before, but I thought it merited a detailed explanation.

My solution lies below...

 Click to enlargen!
I used some artistic interpretation and gave each prince a black or white party hat. Above, the set up is exactly as in the problem. This is from our perspective: We know that we have either a white or black hat, and we know that each of the two other princes is wearing white. I listed out all possible five hats below each diagram and crossed off the obviously known possibilities for our hat.

It doesn't seem like we have enough information to solve the problem. We can only cross off the two white hats, and that leaves three different options for what our hat could be. To solve, we need to think about another scenario:

Let's put ourselves in Prince 1's shoes. If Prince 1 (or any other prince) sees two black hats, he immediately knows that he is wearing a white hat and wins the princess. So obviously the King would not (and hasn't) chosen to use 2 black hats.

This time, Prince 1 sees our white hat and 2's black hat. Because all the princes are very intelligent, Prince 1 has already realized that if his hat was black (2 black hats like the scenario above), we would have jumped up and correctly guessed that our hat is white. Therefore it isn't so simple, and our hat must be white. So, I have an equal chance as Prince 1 here to deduce that our hat must be white.

 Click to enlargen!
So lets revisit the original setup from our perspective. We know that if we have a black hat, Prince 1 or Prince 2 would be able to deduce that their hat is white like above. The other 2 princes are unable to determine an answer. Therefore, the only possible solution is that we have a white hat like the others.

If you have any questions about my wording or the problem, please leave us a comment!
-Tori