I’ll be at Marvelous Math Morning at CJ’s school this Saturday, playing Nim with kids ranging from K-5. One simple goal is to teach them the winning strategy for the version of the game where there’s one pile and each player can draw 1 or 2 chips. I’ve done that with CJ and he really liked it — and I think the *idea* of a perfect strategy is one of those truly deep mathematical concepts that even little kids can grasp.

But what *else* should I do? What other Nims and Nimlikes should I teach these kids and what lessons should I try to impart thereby?

**Update:** First two commenters both mentioned Tic-Tac-Toe. At what age do kids typically learn how to play Tic-Tac-Toe and at what age have they learned a perfect strategy? CJ is in kindergarten and has not seen this, or at least he hasn’t seen it from me. I’ll ask him tonight.

**Update: **Nim a success! I played mostly one-pile, and the kids were definitely able to grasp pretty quickly the idea of winning and losing positions, and the goal of chasing the former and avoiding the latter. I didn’t encounter anyone who’d played nim before. I felt some math was transmitted. Mission accomplished.

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I think two-pile Nim is even easier for people to grasp sometimes than one-pile. Also, just going over Tic-Tac-Toe (or simple variants) carefully helps them see game trees and hence understand what it means for a game to have a winning strategy.

This next one is only fun if you make people play fast. Take an 8×8 chess board. The first player may cross out any square. Players then alternate crossing out squares adjacent to the other player’s most recent move. Last player that is able to cross out a square wins. [The winning strategy is in the next paragraph, so skip it want to think about it yourself.]

The winning strategy: Second player wins. Mentally tile the board with dominoes. The first player’s move fills in half of a domino; fill in the other half yourself. Now, any move by first player fills in half of an empty domino, and your move is to fill it in completely.

Another lovely example are some chase/pursuit games on graphs. For example, question 6 on here:

http://www.mathcamp.org/2008/qquiz.php

I assume they know tic-tac-toe (and thus already know about *that* perfect strategy, though there are still some small surprises, e.g. I learned in an old Gardner column that 1 a2(?) b2 2 c2?? loses against _any_ reply!). Now give them a pile of face-up cards numbered 1 to 9; they draw in turn, and the first to get three cards that sum to 15 wins. [rot13(vfbzbecuvfz)]

On an empty chessboard place White Rook a1, Black Rook h8. Play normally except that if you cross a square controlled by the opponent you lose (so if the Black Rook started out on b2 White would lose immediately).

If some of the kids know how to play chess (as seems likely) you could show them some Zugzwang positions from Elkies’s paper http://library.msri.org/books/Book29/files/elkies.pdf. Once they realize the connection to Nim, they will have learned that there are fun, interesting games where situations can arise in which you would rather it *not* be your turn. This is so even in non-CGT contexts (e.g. http://en.wikipedia.org/wiki/Endplay, or the sprint event in track bicycling).

Come to think of it, in the R vs. R game we could allow only forwards moves (White N/E, Black S/W); the basic strategy doesn’t change, but the game then easily bijects with two-pile Nim. This is even clearer with just one Rook, which starts on the far corner and each player moves in turn West or South aiming to be the one who gets to move the Rook to a1. This introduces the fundamental concept of a two-dimensional state space for a pair of one-dimensional variables. So maybe start with the one-Rook game and two-pile Nim, then build up to the two-Rook game I suggested first.

The one-piece game becomes much more interesting if you replace the Rook by a Queen (which can move SW as well as S and W), but you knew that already from _Winning Ways_, and it’s probably too much to pile on your K-5 charges…

P.S. Thanks to Sam for the plug for my paper. Zugzwang is of course central also to the other games we’ve described other than Tic-Tac-Toe.

Since you ask. For kids on the 5 side of K-5:

1. All impartial games are nim. So it might be nice to show them some other impartial games (eg sprouts, kayles, hackenbush etc.), play them a bit and see why they “are” nim; (instead of partizan games like Tic-Tac-Toe or Chess (fun though those are))

(my eldest daughter is in 5 and loves sprouts)

2. Impartial games (i.e. nim) all have characteristic 2; i.e. it is simple to see that playing two copies of an impartial game is an easy second player win (i.e. game+game=0)

Ask my friend Japheth Wood, who has recently shown nim games to a bunch of people including kids. You can reach him at http://themathwizard.blogspot.com/

One of my favorites when I was a kid was called Screw Your Neighbor, as one teacher made us change the name to, Upset the Person to Your Immediate Left. All you need is a deck of cards. The dealer passes out one card to everyone, including him/herself. The objective is simply to not be the one with the lowest card. Play starts with the player to the dealer’s left. You have the option of either keeping your card, or exchanging it with the person to your left. It continues around until the dealer, who has the option of keeping his/her card or drawing a new one.

Obviously probability is at play here but you have to make some sort of guess at what card is making its way around before deciding to dump off, say, your five of clubs.