## Monday, December 5, 2016

### Leftorvers with 100 game

In Grades 3 and 4, we played a nice game that (I think) we got from Marilyn Burns. Looking for a reference after the fact, I see it explained in her book Lessons for Extending Division.

Basic play

• Start with a target number (we used 100) and collection of available divisors (we used integers 1 to 20)
• Players take turns choosing a divisor from the remaining available options. They divide the current target by that divisor and keep the remainder as their score for the turn. They also subtract the remainder from the target to create a new target for the next player.
• Each divisor gets crossed out when it is used, so it can only be used once.
• The game ends when the target is reduced to 0 or when all available divisors are exhausted.
• We played as a two player game.

Here's an example of a game play:
Player 1 chooses 17. 100 = 17 * 5 + 15, so player 1 scores 15 points, the target is reduced to 85, and 17 is no longer available as a divisor.

Player 2 chooses 20. 85 = 20 * 4 + 5, so player 2 scores 5 points, the target is reduced to 80, and 20 is no longer available as a divisor.

Player 1 chooses 14. 80 = 14 * 5 + 10, so player 1 scores 10 points, the target is reduced to 70, and 14 is no longer available as a divisor.

Player 2 chooses 18. 70 = 18 * 3 + 16, so player 2 scores 16 points, the target is reduced to 54, and 18 is no longer available as a divisor.

Player 1 chooses 19. 54 = 19 * 2 + 16, so player 1 scores 16 points, the target is reduced to 38, and 19 is no longer available as a divisor.

Player 2 chooses 13. 28 = 13 * 2 + 12, so player 2 scores 12 points, the target is reduced to 26, and 13 is no longer available as a divisor.

Player 1 chooses 15. 26 = 15 * 1 + 11, so player 1 scores 11 points, the target is reduced to 15, and 15 is no longer available as a divisor.

Player 2 chooses 16. 15 = 16 * 0 + 15, so player 2 scores 15 points, the target is reduced to 0 and the game ends.

Player one wins 52 to 48.

Our experience
We found this to be a fun, interesting, and engaging game. The practice with dividing and remainders was pretty obvious. In addition, it opened up some opportunities for strategic thinking, particularly at the end-stage of the game. I think there are also several good extension explorations.

Extensions
First, I created a simple pencilcode program for two players to play this game against each other. Here's a playable version (and here's the code).

Second, you'll notice that the first player in our sample game followed a "greedy strategy."  At each stage, that player chose the divisor that would give the most points on that turn. If you look closely, that isn't the best strategy at the end of the game.

So, a natural exploration is to find the best strategy for different starting targets. One specific point of about which we're curious: is it ever desirable to skip your turn (choosing 1 as the divisor is effectively a turn skip)?

Some other areas for investigation:
• must the game always end on 0 or can we run out of divisors?
• given a target and collection of starting divisors, what is the shortest (number of turns) game possible? What is the longest game (number of turns) that does end at 0?