## Tuesday, June 2, 2015

### 31 Game (math class 1 year 2)

who: 2nd and 3rd grade at Baan Pathom Tham school
when: Tuesday morning, around snack time
what did we use: regular playing cards

In our class today, we mostly borrowed the game 31 from Dudeney, via Denise at Let's Play Math. You can find the basic rules there, so I will write about slight variations we played and our experience playing with the kids.

Variations
The first modification we made was to start playing as a group instead of head-to-head. Going around in a circle, each student took a turn choosing a card from the grid and adding that value to the running total. This made the game less competitive and involved more people. Because it was less competitive, the kids were willing to choose cards fairly randomly and speed up game play. This was a big advantage for introducing the game, otherwise a couple of the more competitive kids wanted to take a long time thinking through the game to work out their own strategy.

This version also offered natural points where we could ask the kids to look ahead to see who would be able to reach 31, with everyone participating in the question.

After playing this version for a while, we split up and continued this way with half the class. After another couple of rounds, we split into teams and played head-to-head.

Finally, we added cards 7, 8, and 9 to the grid and played to a target of 61.

*Update*: After talking to one of the parents about this game, we came up with another variation. For this, arrange the cards in order 1 to 6, but turn them face-down. Players then alternate turning the cards up and the running total is the sum of the cards showing. Two reasons for considering this version: first, it provides an extra pattern for the kids to consider when they are deciding which card to turn and, second, it is easier for them to recalculate the running total based on the values they can see than the ones they can't see.

Observations
In the basic strategy for the 31 game, the numbers 24, 17, 10, and 3 have special significance. As we were playing, most of the kids recognized the importance of 24. A couple were also able to work back to 17. I didn't see anyone fully work out the associated strategy. Then, it will be fun to see if they can figure out that this isn't a complete strategy for the game.

When playing the game, the kids who just wanted to play randomly were still very interested in questions that hinted at strategy. For example, I would pause the play briefly to ask whether either one had a winning strategy from that point or for someone to explain why they had made a certain play. Generically, these were good conversations.

The kids who wanted to figure out a strategy were generally less fun as playing partners because they were cautious and slower in their play. I'm not sure whether this is a feature or bug.

In second grade, all of the games ended exactly on 31. In third grade, they managed to get a couple of endings where 31 was impossible. I don't think they were yet able to work out exactly what happened to create that result.

Finally, I am looking forward to discussing the targets. Is 31 a good value for the 1-6 game? What would be different if the target were lower or higher? What about the 1-9 game or 1-10 game? What makes a good target anyway?

Homework
Playing either A (value of 1) to 9 or A through 10, play 5 times and record the outcome. Write down your thoughts on strategy for how to win this game.