We started a new session of enrichment today. This time around, all of the students are receiving the same instruction — they are not divided into math and humanities groups. This means that I see all of the students once each week. We are working on a strategic thinking unit, and these skills are ones that all of the kids will be able to use.
Today we played NIM. There are a number of different versions of NIM, but the one we played starts with counters set up like this:
The object of the game is to “stick” your opponent with the last stick.
In this version, the players take turns removing any number of sticks from a single row.
We played on the Smart Board to begin, with the kids taking turns playing against me. They all quickly figured out “bad” combinations of sticks. For example, you never want to find yourself with three rows of one, or two rows of two.
The students played in pairs for awhile and we stopped frequently for people to explain their strategies. Many kids thought they had the solution — until they played against me at the Smart Board. That’s how it was supposed to work — I knew the secret and they didn’t!
I then told the kids the first secret — you can always win if you go second. If you go first, you can still win, but only if your opponent makes a mistake. Then I explained the strategy in the simplest way I could. (I cribbed this explanation from here. I found it the easiest breakdown to understand.):
The winning strategy is:
You must always take as many matchsticks as possible so that the “Nim sum” of the rows remains ZERO.
What is a “Nim sum”?
Count the matchsticks in each row… And convert them mentally in multiples of 4, 2 and 1. Then, CANCEL pairs of equal multiples, and add what is left. So, when starting, the “Nim sum” of the rows is:
|Row1 = 1||= 1 x 1 = 1||=|
|Row2 = 3||= 1 x 2 + 1 x 1||=|
|Row3 = 5||= 1 x 4 + 1 x 1||=
|Row4 = 7||= 1 x 4 + 1 x 2 + 1 x 1||=
|Total of UNPAIRED multiples||= 0||0||0|
As you can see, there are currently TWO 4’s, TWO 2’s, and FOUR 1’s (= TWO + TWO + FOUR = 8). You have then an EVEN number of multiples, the remainder after dividing this number (8) by 2 gives 0.
To win at Nim-game, always make a move, whenever possible, that leaves a configuration with a ZERO “Nim sum”, that is with ZERO unpaired multiple(s) of 4, 2 or 1. Otherwise, your opponent has the advantage, and you have to depend on his/her committing an error in order to win.
We played several times on the Smart Board using this strategy. Some of them got it; some of them didn’t. They all became much, much better at anticipating what their opponent would do and what would happen after that, and that’s really what I was hoping for.
You can play NIM for free online. Not that I do that in my free time or anything. Find one game here.