Friday, December 21, 2012

The Prediction

You might have seen this before, for example in this video, or on a TV show. While in the video the tiles that are removed are of course always the same, in the above game the removed tiles and the number of places to move are randomized — and yet it still works. But how?

Let's number the tiles and color them like a checkers board as follows:

The “no diagonal moves”-rule gives us quite some knowledge. For example, if you start on tile 2 and then move an odd number of places, you will never be able to end up back on 2 where you started. In fact, you won't be able to end up on any of the even (black) numbers. The same goes for any even (black) number you start on. Try it!

It works the other way around too. If you start on an odd (white) number, and move an odd number of places, you will always end up on an even (black) number.

Let's put it like this: When moving an odd number of places on the checkers board, you will always change colors. When moving an even number, you will always keep the same color. This makes perfect sense.

So, the prediction game above ‘knows’ what your starting tile is (is it black or white?) and it ‘knows’ the number of places you will move (will you keep your color or switch?). Throughout the entire game it keeps track of which color you are on, and thus each turn it can safely remove a random tile from the opposite color.
At the end there will be only one tile left. This tile is more or less random, but of course the game pretends it knew all along...

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