Nim: A Series of Number Games

The purposes are the same as in the game of Krypto:

1. 1. To build interest in mathematical thinking
2. 2. To develop skills in mathematical thinking
3. 3. To provide practice in developing and using cognitive strategies
4. 4. To provide practice in counting, addition, subtraction, multiplication, and division in a context other than pure drill

I. Counting Practice

1. Two people count alternately, starting with 1.

2. The one who gets to 20 wins.

3. The sequence goes like this: (The first person's numbers are blue; the second person's numbers are red.)

	First	Second
	1	2
	3	4
	5	6
	7	8
	9	10
	11	12
	13	14
	15	16
	17	18
	19	20	Winner!

4. As you can see, the blue choices are all odd, and the red choices are all even. Thereore, the first player has no chance to win this game. However, if you select 21 as the winning number, the second player has no chance to win. This is mathematical thinking that young children can develop through games like this. Once this is understood, of course, this is no game at all. However, as a learning tool, these two comments should be considered:

1. a. The game, in its simplest form, can be used as counting practice in the Kindergarten and the first grade.
2. b. the game can be extended in many different directions, as illustrated by the adjustments shown below, and used at many different grade levels.

5. Question: If the two players count by twos, and the winning number is 14, which player will win, the one who goes first or the one who goes second?

6. Question: If the two players count by twos starting with 1, and the winning number is 17, which player will win?

II. First Adjustment (Addition Practice)

1. 1. Connection: Adding is a short way to count. The introductory game above, counting by ones, can just as well be thought of as adding 1 each turn to an accumulated sum. Now, adjust the introductory game so that each player has the option each turn of adding either 1 or 2 to the accumulated sum. (This is equivalent to saying each player can count up either 1 or 2 from where the other player stops.) If the winning number is 20, the first player can always win if that player chooses the correct number at each turn. If you are the first player, choose the blue numbers in the table below, and the second player cannot possibly win.

	First Player	Second Player
	2	3 or 4
	5	6 or 7
	8	9 or 10
	11	12 or 13
	14	15 or 16
	17	18 or 19
Winner!	20

1. 2. Imagine that the winning sum for the above game is 21 instead of 20. Then you could be absolutely sure to win only by going second. What sequence of numbers would you get as the second player to be certain to win?

   3. It would be a much shorter game if the winning sum is 10. Would you go first or second to be sure to win this game? Each player can add either 1 or 2 each move. What sequence of numbers would you get to be absolutely certain to win?

    

III. Second Adjustment (Strategy Development)

1. 1. You were given the sequence of winning numbers in the above problem (I think we can call it a problem as well as a game.). How was this winning sequence determined? One is a series of logical statements based on observations, as follows:
   a. I can get 20 if I get 17. (Why? Well, if I get 17, my opponent has only the choice of getting either 18 or 19. In either case, I can get 20 on my next move. This begins a chain of mathematical reasoning.

   b. The next step is to realize that if I can get 14, I can get 17. Why?

   c. Next, if I can get 11, I can get 14.

   d. If I can get 8, I can get 11.

   e. If I can get 5, I can get 8.

   f. If I can get 2, I can get 5.

   g. Finally, I realize that the only way I can be certain to get 2 is to go first.

   h. Thus, to be absolutely certain to win this game is to go first and get the sequence 2, 5, 8. 11, 14, 17, and 20. (Note: It is possible to win this game even if you do not get this exact sequence. But that can only happen if your opponent does not know the secret to be absolutely sure to win.)

   2. The above strategy is called Working Backwards and is quite useful in many mathematical problems and proofs. Use the strategy of Working Backwards to list the series of numbers you would have to get to win game in which the winning sum is 100, and the number each player each turn can add any number from 1 through 10.

   3. Make up a game with a winning sum between 39 and 50 in which each player each turn can add any number from 1 through 7, and which you would have to go second to be absolutely certain to win.

IV. Third Adjustment (Subtraction Practice)

1. 1. The Working Backward strategy applied to the game in III. 1, can be though of as subtracting 3 (1 more than the maximum amount that can be added each turn) from 20, and then continuing to subtract 3 until you reach a number less than 3. The number you obtain in this way is the number you want to start with. Of course, if the number you reach is 0, you must go second to be sure to win.

   2. The use of the Working Backward strategy in this way provides some practice of subtraction in a different context.

   3. Another adjustment to provide practice in subtraction is to have a game like this one: The game starts with, say 30, and each player can subract in turn any number from 1 through 7. The player who gets 0 wins. What would you have to do to be absolutely certain to win this game? This is a good game played on a calculator. The calculator is passed back and forth between the two players between turns.

V. Fourth Adjustment (Multiplication and Division Practice)

1. 1. When you apply the Working Backwards strategy by subtracting the number that is one more than the maximum that can be added each turn, you are using one kind of interpretation of division. The number of times you subtract is the quoient in a division problem. For example, if you are subtracting 11s from 100 until you get a number less than 11, this tells you how many 11s are in 100. The number that is left is the remainder after subtracting out all the 11s. This illustrates the important connection between division and subtraction. Division can be thought of as repeated subtraction. If the remainder is 0, this means that the winning number is a multiple of the divisor. The connection is completed by this statement: The quotient multiplied by the divisor plus the remainder is equal to the dividend.

   2. Use the above connections to calculate the number to start with in a game in which the winning sum is 287 and any number from 1 through 16. How would you describe the series on numbers you must get to be absolutely sure to win this game?

VI. Fifth Adjustment (Creativity)

Make up a similar game, but with some variation. This gives children the opportunituy to use their imaginations. One example would be to add only odd numbers 1, 3, 5, 7, 9, and 11 to get 100. What would be your strategy?