### Brain Teasers

# The Egg Game

There are 2008 eggs lying in front of you. I am also with you waiting to start the game. The goal of the game is to be the person to take that very last egg (or eggs) to win. You and I can each only take between 1 and 100 eggs at a time. You go first. How many eggs must you take initially in order to always win (assuming we both play perfectly)?

### Hint

By a perfect game, that means over any two turns, the sum of the number of eggs you and I take is the same (except of course, excluding the initial move you make).### Answer

89 eggs.In order to see this, look at the smallest way possible to guarantee a win; you leave me with 101 eggs. If I take 1 egg, you take all 100; if I take 100 eggs, you take the last one. Therefore you would always win.

If you try with 202, I can take from 1 to 100, so there is 201 to 102 left. Then you would take 100 or 1 egg(s) in order to leave me with 101 again. By simple analysis, you must take 101 eggs over two rounds in order to get down to 101.

19*101 = 1919, so we must take 2008-1919 = 89 eggs to start off with a multiple of 101.

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