Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 decade ago

Work in pairs on the following version of a game called NIM. A nongraphing calculator is needed for each pair.?

36.

Work in pairs on the following version of a game called NIM. A nongraphing calculator is needed for each pair.

A.Player 1 presses 1 and + or 2 and +. Player 2 does the same. The players take turns until the target number of 21 is reached. The first player to make the display read 21 is the winner. Determine a strategy for deciding who always wins.

B. Try a game of NIM using the digits 1, 2, 3, and 4, with a target number of 104. The first player to reach 104 wins. What is the winning strategy?

C.Try a game of NIM using the digits 3, 5, and 7, with a target number of 73. The first player to exceed 73 loses. What is the winning strategy?

D.Now play Reverse NIM with the keys 1 and 2. Instead of + , use - . Put 21 on the display. Let the target number be 0. Determine a strategy for winning Reverse NIM.

E.Try Reverse NIM using the digits 1, 2, and 3 and starting with 24 on the display. The target number is 0. What is the winning strategy?

F.Try Reverse NIM using the digits 3, 5, and 7 and starting with 73 on the display. The first player to display a negative number loses. What is the winning strategy?

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  • 1 decade ago
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    A. You want to force your opponent to leave either a 20 or a 19 for you. To do that you want to give him an 18. You can get to an 18 if your opponent leaves you either a 17 or 16. To force him to give you either a 17 or 16, try to leave him with a 15.

    See the pattern? You always want to give your opponent a multiple of 3. If he adds a 1, you add a 2 (and vice versa) to get to the next multiple of 3. If both players know this strategy, whoever goes first (adding either 1 or 2 to the initial 0) will lose.

    B. Is similar except that you want to force your opponent to give you a number in the range 100-103. To do that give him a 99, 94, 89, 84, 79, ... any number that ends in a 4 or 9. What ever your opponent does, just bring it back around to a 4 or 9.

    C. You want to give you opponent a 73, 72 or 71, so that he has to go over. The trick here is that with digits 3, 5 and 7, the magic multiplier is 10. So always try to bring the total to a number that ends in a 1, 2 or 3. If you get to start at 0, give him a 3. After that always bring it back around to a 3.

    Now that you see the general strategy, I'm sure you can figure out the rest.

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