Nim is a mathematical game for two players. It is a member of the class of games known as "impartial" i.e., those games where from any given position, the same moves are legal for both players. The rules of Nim are that you start with a number of piles of counters - three piles of 3, 4 and 5 counters is good for beginners - and the players take turns removing any number of counters from one pile. The winner is the player who takes the last counter. (In misere Nim, the winner is the player who forces his/her opponent to take the last counter.)

The strategy for winning Nim is to try to leave your opponent in a position where if you write the number of counters in each pile in binary and add without carrying, the sum of each column will be even. Here's an example of how you work this out with piles of 4, 5, and 6 counters:

```   4    100
5    101
6   +110
---
311
```
Here you have to change the parity of all the columns. One way to do this is to remove one counter from the pile of 4.
Nim is a simple game that encourages logical thinking. I played many variations of this game with my sons when they were in Kindergarten and we called it Take Away, which is what kids that age understand as subtraction. Later on while teaching fifth grade I ran into a way to adapt this game for upper elementary student as a counting game. Finally there is Three Row Nim, the hardest version. It was elating to find an article about it in the September/October 2001 issue of Creative Classroom magazine.

The game is thought to be quite ancient and may have originated in China. In 1901 a full analyses along with a description of a mathematical strategy for playing the game was published by Charles Leonard Bouton. It is Bouton, by the way who is responsible for the modern name, Nim and the name is thought to be German, from nimm the German word for "take". Nim is interesting because every position can be classified as a winning or a losing position. From a winning position, a player can only create a losing position for his opponent, no matter how many counters he removes. However, from a losing position, it is possible to move to either a winning or a losing position.

This is a great game to play if your traveling with kids because just about anything from coins to beans can be used as counters. But keep a sharp eye on the little ones and the beans. Most math games are a fun way to practice basic math skills and are valuable reinforcements. It's the many variations of Nim that makes it so challenging for all ages. If the player wants to win, and they will want to win, they will have to think mathematically!

## The Bean Game

Ages 5 to 9

It's best to use counters such as lima beans colored with markers when playing with young children. Start by placing 12 beans in a row in front of you and explain that you are going to teach them about a game called Take Away, where the person to take away the last bean is the winner. Show them the rules by playing a few games while the children follow along. Here are some simple rules that children this age will understand:

How to play the Bean Game

1. Nim is a two person game where each player has a turn and neither one may pass.
2. On his or her turn, the player has to take away one, two or three beans moving from left to right.
3. Whoever takes the last bean by itself or in a group, is the winner of the Bean Game.
Easy enough right? Well there's more than meets the eye. You can challenge the children to think harder and learn more if you do one important thing-- learn how to win at Nim as frequently as you can. But don't tell the children the winning strategy. Leave it up to them to figure it out on their own!

How to win

Always go second in this game of Nim if possible.
Always look at the 12 beans as three groups of four beans.
Always, if possible, take away the number of beans that leaves a multiple of four.
If you have to go first, or you can't get a multiple of four on your turn, take away a small number of beans. Then wait until you can leave a multiple of four.
By winning as often as you can, you will challenge children to figure out the game of Nim for themselves, and believe me they Will figure it out. When they do, it's time to switch strategies a bit. Try changing the row to 15. Using exactly the same rules. You can go first and by using the same scheme you'll start winning again, and the children will start thinking even harder!

## The Counting Game

Ages 10 to 12

This version is great for the older children. There's no need for any preparations or materials and it can be played at a moments notice.

How to play the Counting Game

1. This version is also played by two people taking turns, but in this game they take turns by counting out loud from 1 to 20 in place of using beans.
2. The person who goes first begins by counting 1, or 1, 2 or 1, 2, 3. The second person then begins to count aloud one, two or three more numbers in order where the first person stopped.
3. Whoever says the number 20 by itself or in a group of numbers is a winner.
For example: 1, 2, 3 (player one), 4 (player two), 5, 6 (player one), 7, 8 (player two), 9, 10 , 11 (player one), 12 (player two), 13, 14 (player one), 15, 16 (Player two), 17, (player one), 18, 19 20! (player two). Player two is the winner.
Again make sure you know the strategy to win so the children will be challenged to play you until they can beat you! The way to win this one is like the Bean Game, but again let the child figure it out on his or her own!

How to win

• Go second every time you can in the Counting Game, if possible.
• Always listen for the multiples of four; if your challenger doesn't say a multiple of four make sure that the last number you say is a multiple of four.
• If you go first or if you can't get to a multiple of four, count on a small number and wait until you can count to a multiple of four.
To challenge the kids even more, add some variety by changing the number to 21, or any number for that matter! But first make sure you've figured out the winning strategies.

## Three Row Nim

Ages 13 and older

By now they are ready for a more challenging game with bigger numbers and more variation. One way to keep the game more defying for older kids is to change the number of beans for the Bean Game of the final number of the Counting Game each time they play.

Three Row Nim is the most difficult version of nim. While it's also a take away game it has small yet major differences. It begins with three rows instead of one-- a row of three, a row of five and a row of seven.

How to play Three Row Nim

1. This again is played by two people taking turns. No passing on turns and they have to take away at least one bean.
2. During their turn, the player can take away as many beans as they decide, but only from one row.
3. The player who takes away the last bean, either by itself or in a group of beans is the winner.
Unfortunately, the winning strategy is above my head for explaining but it would make a nice write up challenge for the more mathematically inclined noder. Here is a hint: It involves binary numbers. If your children have figured out how to win the Bean and Counting Games then they will be up to this challenge of figuring out how to win Three Row Nim. You can encourage them to keep records of their methods and the amounts they and their partner chose each time and track which person who won. For example, if you are left with two rows that have an equal number of beans in them, and it's your turn, the other person simply has to match your move in the other two rows each time to keep the two rows even and you will lose. Have them test out thier methods by trial and error keeping a record to see which techniques work. Good Luck!

Sources

Nim resources:
www.spdcc.com:8431/nim.html

Paul Giganti, Jr., The Game of Nim, A simple way to encourage logical thinking, Creative Classroom (Sept/Oct 2001), 76.

Note: Paul Gianti, Jr., provides mathematical development for K-8 teachers in the US and is the author of many children's books, including How Many Snails? (Greenwillow, 1988).

In older writings, from about the 600s to the 1600s, nim, nime, or nym was one form of the verb meaning 'to take', and its many related senses. It comes from the West Germanic root nem-, and in the days of Old and Middle English nim and the Scandinavian-rooted tacan battled it out over which would be the English verb of choice. Obviously, 'take' eventually won out.

In Old English niman meant to take, accept, receive, grasp, pick up, catch, or steal. At that point it was fully conjugable, and nim was only the imperative, first person, and second person subjunctive forms (If you would like to see the full set of conjugations, I recommend this site). Over time any number of conjugations and alternate spelling were used, including a number of nom, noom, and nome variants. Forms such as neme and nomen might easily be confused by the casual reader as variants of 'name' or Latin 'nomen', which is why there are so very few casual readers of Old English.

The use of nim decreased precipitously during the 1500s, and by the early 1600s the primary remaining usage was in the sense of 'to steal'. At this time is was also used as a noun; a 'nim' was thief. This too has dropped out of usage, and I am not aware of any words in current usage that derive from the root nim.

A particularly well-read person might also find nim used as a name for the Indian neem or margosa tree.

References:
The University of Michigan Middle English Dictionary
Etymology online
The Oxford English Dictionary

Nim (?), v. t. [imp. Nam (?) or Nimmed (); p. p. Nomen (?) or Nome ().] [AS. niman. &root; 7. Cf. Nimble.]

To take; to steal; to filch.

[Obs.]

This canon it in his hand nam. Chaucer.

[Editor's note: Compare with the German word nehmen, meaning "to take". Its familiar imperative form is "Nimm!"]

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