Puzzle: Infinite Hats Problem

February 24, 2014 § 2 Comments

For those of you who are unaware, I have been in Budapest for a little over a month, and I have been quite a bit busier than I expected. I know I have been much worse than I had intended about keeping up with this blog, but I heard a great puzzle the other day, and it is quite exciting. Here it is:

There are infinitely many logicians (one for each integer), and an adversary tells the logicians that he will place a hat on each of their heads. He tells each logician that each hat will either be red or blue. He then tells each logician that, once he has placed a hat on each of their heads, the logicians may see the color of all the hats that are not their own, and then they must attempt to guess their own hat color. The logicians win if only a finite number of them guess incorrectly. Is there a strategy that will allow the logicians to win if the adversary knows their strategy?

Just to make sure the problem is clear, each logician gets a different hat. At no point do they get to see their own hat color nor do they receive any information from any of the other logicians once they have received their hat. All they can see is the color of the hat placed on each other logician’s head.

Thanks for reading!

-A Student of Logic

Puzzle: Catch the Shark

December 1, 2013 § 2 Comments

I was speaking with my editor last night, asking for advice on something quick to write about. After running through a list of drafts and ideas that I had written and determining that none of them seemed to fit what we were looking for, he started reciting to me a list of what I will refer to as “cocktail party” pieces of Mathematics. After hearing a few items, I find myself saying “that reminds me of that problem where…” to which he says “that problem where you are trying to catch the shark in the boat?”

Of course, he was correct. The problem it reminded me of was one which I had heard him and a mutual friend recite quite a long time ago, and it was also exactly what I was looking for. The problem is as follows:

You are the captain of a boat. This boat is on a one-dimensional body of water, and this body of water goes on forever in both directions. To elaborate, each possible location on the water is an integer point, so if you are at location 2, and you go to the location immediately to your right, you will be at location 3. If you are at location 0, and you go to the location immediately to your left, you will be at -1. In general, if you are at location and you go to the location immediately to your right, you will be at location x+1, and if you instead go to your left, you will be at location x-1.

Your boat is special, however. You are not restricted to just moving to locations immediately to your left or right. You can enter an integer number into your boat’s navigation system, and at the next second, you will be at that location. For instance, if you are at location 3 and then you enter into your computer-1036, at the beginning of the next second, you will be at location -1036.

Now that you know how the boat works, here is the challenge. There is a shark in the water. You know he is somewhere in the water, but you have no idea where he is. Furthermore, he is moving at a constant rate through the water (the rate is some integer distance per second), but you also have no idea what the rate is. Can you catch the shark (that is, can you, put yourself at the same location as the shark)?

To recap the problem: You are in a boat. At each second, you can pick a new location to be in (you can think of this as being able to take a guess at where the shark is once every second). The shark moves at a constant rate, and you have no idea where he started. Can he be caught?

As always, feel free to message me with a solution or to ask for a solution. Try not to spoil it for others.

Thanks for reading!

-A Student of Logic

Puzzle: Magic Square

October 29, 2013 § 7 Comments

Hello Math people. I heard this puzzle a few weeks ago, and I thought it was pretty great. Here is how it goes:

You and a friend are playing a game against an adversary. The game is played as follows. You walk into a room with the adversary, leaving your friend outside. The room contains a checker board (the typical 8 by 8 kind), and each square has on it exactly one coin. Each coin can either be heads-up or tails-up. Then, the adversary chooses exactly one square which he calls the magic square. You, then, based on the arrangement of the coins and which square is the magic square, will choose exactly one square and turn its coin over (either from heads-up to tails-up or from tails-up to heads-up). Then, you will leave the room through a back door and wait in a separate room. Your partner then comes into the adversary’s room, looks at the board, and proceeds to take a guess at which square is the magic square. If your partner guesses correctly, you win.

Assuming you and your partner have no idea of what arrangement the board will be in when you enter the room, come up with a strategy for you and your partner to use so that you can always win.

Note: the only information you or your partner can have from looking at a coin is which square it is on and whether or not it is heads-up. The solution does not involve doing something silly by rotating the coins; the adversary chooses the orientation of the coin after you flip it.

Thanks for reading, and feel free to message me if you would like the answer!

-A Student of Logic

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