I’ve been taking some time to write up some lessons, and I’d love to get some feedback. You can click here for a pdf of this lesson on 1-2 Nim. It’s one of our favorites: a surefire way to get students of any age playing and thinking.

The question is: how’s the write up? We’ve been shooting for 1-2 pages maximum, so we keep it streamlined and easy to use. Teahcers, is this a usable format for you? Parents, do you feel ready to try this out with your kids?

Anyone who tries this lesson out with a child, student, or class, please let us know how it went in the comments. Any constructive feedback is welcome!

Thanks!

Dan,

The write up looks good to me. Nim is a great game to get kids to start thinking. I have a similar write up that I could share if you’re interested. I think the biggest issue teachers or parents face is trying to help the kids reason for themselves vs. just telling them how it works. The table you’ve created will certainly help with that. Is the intent to give students the table right away, or after they’ve played for a while? Depending on the ages of the students, it might be interesting to see if they could develop the table on their own.

Here are some questions that we’ve documented which have helped students:

“Would it help to start with a simpler situation?”

“What happens if you leave just one or two tokens at the end of your turn?”

“What are the possibilities if you leave three tokens at the end of your turn?”

Hope that was helpful. Thanks for sharing.

Hi Raj,

Thanks for this feedback. The intent is not to give the table to students right away, and maybe that’s something I need to make clearer here. I like your questions as well–that’s something we do when we teach, and maybe having a place on the sheet of good questions for the teacher to ask would be another good thing to add.

I like the idea of some Polya-style questions included in writeups. Starting with the simplest situation and working your way up is such an important technique.

Love Nim. Totally agree that the greatest challenge is supporting kids in “figuring out” the game without just telling them. A prompt I’ve found helpful is “Let me know when you know who is going to win.” My only other suggestion is to be really patient (after all, if the game is still perplexing to the student because they haven’t solved it, they’ll want to keep playing).

And a few more variations, just for kicks…

1. You’re allowed to remove 2, 3, or 4 tokens (whoever *is the last to take a token* wins).

2. Play with more than 2 people where the last person to take a token wins, the 2nd to last person to take a token gets 2nd place, etc.

3. You are allowed to take any Fibonacci (1, 2, 3, 5, 8, 13, …) number of tokens. (Also nice with powers of 2)

One of the things I’m struggling with on these is how much to reveal of the solution. I’d like adults who use this to have the experience of working through the problem themselves; but I also want them to feel secure enough that they’re willing to. I imagine that there may need to be some redundancy when it comes to helping the leaders not give away too much. Emphasizing patience may be the right way.

I love the Fibonacci number variation, by the way. Never seen that one before.

Hi Dan & Katherine,

I played a variety of Nim games with 20 4th & 5th graders at math club a few weeks ago. I had about 30-45 minutes of good engagement. The biggest challenges I had were a) matching players at appropriate levels in an after-school setting b) feeding them the next level (more piles, bigger totals, counting up, counting down, last player wins/loses) at just the right rate. As a parent of a 5th grader we got about two more hours of play time at dinner that week (at his urging, not mine)

Here are some thoughts on the lesson plan.

a. Mention an initial time-frame.

b. Connect it to other commonly known solved games, e.g. tic-tac-toe & chop-sticks.

c. Suggest a ladder of difficulty based on your experience, and possible entry points based on age/grade.

d. Give teachers strategies to mix up the players (teams, tournaments, etc.)

e. Suggest a “if you liked this, then try ….” e.g . Mancala.

f. I like the idea of a list of questions that teachers can ask. Could you frame it as “buy a hint.”

e. Finally, in terms of giving students strategies to approach challenging problems, name the strategies used in solving Nim games.

Thanks so much for sharing these, and so many other lovely puzzles.