## Thursday, July 10, 2014

### NIM

Who: J1 (7 years old) and daddy
When: just before bedtime, game lasted about 10 minutes
Where: on the floor
How: using pattern blocks (though any collection of small objects would have worked)
What: playing NIM (rules below), counting blocks, making array patterns with blocks
Why: finding some one-on-one time, adding another strategy game and building from this game we played before, reinforcing number sense.

Following a good suggestion, I'm trying a new format with these notes.  Instead of just saying what we did, I'm going to try to give more of a recipe so that it will be easy (easier?) for all y'all to play along at home.  I'm giving the ages of the children involved as a rough guide to whether your kids might have a similar experience, but I hope you don't take that too seriously because:
- age is not a very good proxy for experience, sophistication, or interest
- most of the things we do are low threshold (almost anyone can get started) and high ceiling (almost anyone could find an interesting and challenging extension to the basic activity).

NIM rules:
split a bunch of little objects into several piles.  two players take turns removing as many objects as they want on their turn with two conditions: you can only take from one pile at a time and you have to take at least one object.

What we did
We played with the pattern blocks and didn't actually separate into piles. Instead, our rule was that you could only take one type of block at a time.

The first time, I poured out about 1/3 of the blocks and separated out the hexagons, trapezoids, and triangles.  We played one game that way, basically carelessly taking away various amounts for a while. I took away all the green triangles at one point to make the strategic positioning a little clearer.  When we got to a small number of blocks (about 5 -7 of each type), I started pausing before my moves which signaled J1 to start thinking ahead and considering strategy. We got to a configuration 1-4 on my move and his eyes narrowed when I reduced it to 1-1 as he saw that I was now in a winning position.

We played again with the other 3 types from the original pour.  Same general method of play, but J1 started thinking strategically a little earlier in the game.

Finally, we poured out the whole bucket and played with all 6 types.  Generally, same mode of play without extensive strategic analysis.

whenever I took away blocks, I would arrange them in a grid of some form and state a multiplication equation associated with that array.  For examples:

• 12 green triangles: I arranged them into 4 x 3 and said "4 groups of 3 is 12"
• 5 yellow hexagons: I arranged them into a 1x5 line and said "1 group of 5 is 5"
• When J1 took away 9 orange squares, I asked if he could organize them and he made a 3x3 square.
Next stage/Extensions
Analyzing the strategy is an obvious next step. For us, I can build off J1's obvious calculations toward the end of the game in this way:
"Hey, when we got to 2 shapes, I noticed you thinking more carefully.  What were you analyzing?"
<discussion>
"What if we start with just 5 triangles, what would you do?" "9 triangles, 15 triangles, oh, its all kind of the same."
"what about 2 triangles and 1 square"
"can we make a list of when you know that you would win or when you know that I would win?"
"what patterns do you notice?"

There are a lot of possible variations and I think it is a great fun for the kids to think of alternatives and see which ones are interesting to play. Two alternatives that came up most readily when we played were misere (last player loses) and a points game where you get points for every block you pick up and the last player gets a bonus.

If you want to take it in a different direction, help them code a version of the game in your favorite programming language. Maybe pencilcode?

To involve other sense, play the game using small pieces of food that get eaten on each move. Or you can create a variation on your favorite musical instrument (say moving from left to right on a piano where every move has to be within an octave).

What do you learn?
Working back from the answer and building up from smaller examples are two generic strategies that you can practice in analyzing this game. This is also a place to introduce recursion or induction.

#### 1 comment:

1. Mathbabe has a great post on Nim. In particular, I am going to have to do the rook example, since the Js also like chess:
Nim at HCSSIM