# Imp Linear Nim Pow

IMP

POW Linear Nim

In this POW, we had to play a game called Linear Nim. In this game, we drew 10 lines on a paper, and we had to take turns crossing out 1, 2, or 3 of the marks. The person that crossed out the last mark was the winner. The first task of this POW was to find a winning strategy for this game. After we found this out, we were supposed to make variations to the game, for instance starting with more or less marks, or allowing a player to cross out more or less marks. We were supposed to consider a variety of examples and look for generalizations in strategies. When I first played the game, I didn’t really have any initial strategy in mind. I just picked numbers at random to choose until getting down to the last few. I thought this game was very simple and easy to understand. One key insight I had was to cross out three marks to get rid of most of the marks. When it came down to the final three or four marks, I had to look carefully at them to see what numbers I could choose before the next person’s turn, so after their turn there would be one, two, or three marks left. For instance, if there were 5 marks left, I could cross out one, and then any number the next person picked would leave me with a number of marks I could cross out with one, two, or three crosses. This is when I realized the strategy of this game. The strategy I developed for the original game was that to win, you have to cross out a number of marks so that there are four marks left before the next persons turn. If your opponent picks three, there will be one mark left, and you can cross it out and win. If your opponent crosses out two marks, there will be two marks left, and you can cross those out and win. If your opponent crosses out only one mark, there will be three marks left, and you can cross those out and win. If you get down to four marks before the next person’s turn, there is no possible way you can lose. At least without any variations. My first variation was...

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