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

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