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Strategy

- If player one chooses to take away 3, then player 2 will win. If player one chooses 2, player one will win. If player one chooses one the outcome is unknown. The goal of the game is to either go first, but if you can’t hope P1 chooses 3 or one.
- When you play 15 nim, your objective would be to get to ten. When you get to ten play with the strategy for 10 nim. If P1 chooses 3 choose 2 to get to ten. If he chooses 3 choose 2 to get to ten. If P1 chooses 1 choose 1 so they can get to ten or you can take another turn to get to ten. You can also get to nine to win. If you get 9 in a scenario, heres what to do. If P1 chooses 2 choose 3 to get to four so you can win. If P1 chooses 3 choose 2 so you can also get to four. If P1 chooses 1 then the outcome is again unknown.
- In any game more than 10 nim, such as 20 nim or 100 nim, the strategy would be get down to ten ASAP. Once you get to ten you can play with the strategy for 10 nim as previously stated.
- If the maximum for each turn was changed to 2, 4 or 6, three things would happen
- If P1 chooses 2 (which is now the maximum) then keeps taking away two then he would win. But, if there were struggle from the opposing player, then the strategy would to play one or two. Lets say P1 takes away 2 but P2 takes away 1, then you can take away 1 to get back to 6. You would also want to be the one to have the last 2 or one ticks.
- If P1 chooses 4 (which is now the maximum) you could take away 1, for the slim chance he would choose 4 again you could choose. thinking that he would not do this, you can still take away one, because if he got down to four then either player could win. so the objective would be to get to five, so you can let them take away how many they want, then just take away the remaining ticks (their will probably be either 1-4 ticks left)
- If P1 chooses 6 (which is now the maximum) then you could choose 4 easily and win.
- If P1 chooses 5 then no matter what you can win. you can take away 5 then win.
- But this is only for ten nim, what if the number were higher? lets take 30 nim, what would happen?
- If the maximum were 2 then you could take ten turns (5 each) to get down to ten. But if either player chooses one it could screw up the pattern and either (defensively) screw the player up or (offensively) give them the upper hand. The objective would be to get to ten and play with the winning strategy for P1 for ten nim.
- If the maximum were 4 (again) then you could each take four away until you got to ten, so a total of five turns. But the other player will probably not do this, because he must keep in mind he has to get down to ten to win. If you get to ten, again play with the P1 winning strategy for ten nim. If the player (1 or 2) decides he wants to mess up the opposing player then that would be either a great (offensive) or a bad(defensive) . All you need to do for now is focus on making the other player to choose ten so that would be where you start.

Thought Process

When I first started this investigation, I didn't know what to do. After a few rounds of playing with the other player, I came up with this conclusion: the person who goes first always wins. After a few rounds I found out this theory was partially incorrect, and I just sucked at the game. But a question arose: Is there a way to combat player one successfully? As the game went on I decided to see what happens as I went first. I chose to take away three, later I lost. I took away two next time, and I won. I took away one and the other player won. I took away one again but I won that time. So the strategy became: If P1 chooses 3 P2 can win, If P1 chooses 2 P2 loses, If P1 chooses 1 the outcome is unknown.

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Generalization

While playing nim, I found this pattern. 3 loses, 2 wins, 1 is an unknown outcome. Its like that saying “First is the worst, second is the best…” Though this only applys to ten nim. So now knowing this strategy, you know its best to go first and choose 2 so you can win every time.

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Evaluation

I did find this process educational, because it made me think about this game harder than I would have. If I didn't have to create a strategy I would have just gone off of luck. I liked this problem and i feel better prepared if i have to do anything like this again. On the other hand it was very difficult and made me think, but the entire problem was still fun.

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