91 – the prisoner and four herrings

Problems with a Point is a great website!  You can search for problems by TOPIC, HABITS OF MIND (STRATEGIES), MATHEMATICS BACKGROUND, TECHNOLOGY, or DURATION. “Open the Lock” is from the site, and I really enjoyed working on it (albeit there was hair pulling).  It’s challenging for my geometry kids, but the lively and thoughtful discussions that come out of the groups make it worth presenting.

Open the Lock

A prisoner is left alone for a few minutes, and tries to set himself free. The lock of his cell is a cylinder separated into four vertical sections. Each section holds a herring, which is oriented either tail-down or tail-up: The lock opens when all four fishes are oriented the same way.

The prisoner can reach the lock, but, unfortunately, from the inside he cannot see all the herrings at the same time. He can only reach any two herrings at a time, take them out, look at them, and put them back (having changed the position of one of them or both or just leaving them as they were). After that, either the door opens, or the lock spins for awhile so fast that it is impossible to keep track of which fishes were touched.

Of course, the captive can set himself free by pure luck, after he changes the orientation of herrings randomly. The question is: can he work out an algorithm to open the lock for sure? If the answer is yes, create this algorithm. If no, explain why.

This year with the whiteboards, this problem is even more fun!

91-3

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