1. Alice and Bob play the following game.
• Initially, there are 20 numbers:10 numbers 1 and 10 numbers 2.
• On each step one of the players select two numbers; and if they were the same, replace them by 2; otherwise, replace them by 1.
• Alice make the first move and they do moves one after another. Alice wins if the last number is 1 and Bob wins if the last number is 2. Who is the winner? Note that this game is not a combinatorial game
2. In the subtraction game where players may subtract 1, 2 or 5 chips on their turn, identify the N and P positions.
3. Is the Nim position (1,3,5) an N-position (explain your answer)?
4. Consider the Misere subtraction game where players may subtract 1, 5 or 6 chips on their turn, identify the N and P positions.

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1. A winning situation for Alice is when there is only one 1 left. Then no matter how many 2s are left, any further move cannot remove the last 1. However, the number of 1s can only be changed by two: take two 1s and put down 2. Therefore the number of 1s remains always even, so Alice must lose and Bob must win in the end, no matter what Alice does...

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