Each partner duplicates their ring – possibly rotating it so the zeros are not lined up. So let’s say I’m playing against you. I make only the three negative: 0, +1, +2, -3, +4.

I rotate my inner ring as much as I want so the zeros are not coupled. Here the inner ring is rotated one notch… The product of the number couples are:

(1)x(2) = 2

(2)x(-3) = -6

(-3)x(4) = -12

I add these. The result is -16.

So I get a score of 16 (the absolute value of -16). Whoever gets the lowest score wins. Did I make a wise choice by rotating one notch clockwise?

No! If I had rotated two notches clockwise, I would have done better:

(4)x(1) = 4

(1)x(-3) = -3

(2)x(4) = 8

Adding these up gives 9. I would have gotten a score of 9. Lower scores are better.

Here are all the results I could have obtained. Some of your students will understand the symmetry or rotating clockwise and counterclockwise.

Anyway – I got a score of 16. Let’s see how you did.

You chose to make a ring: 0, -1, -2, -3, +4. How far did you want to turn it?

If you turned it two counter-clockwise you’d score 9…

…but you could have beaten me by more by just rotating it by one notch. Students should be given one minute to create their rings and another one minute to rotate them. Students may need different amounts of time to do the multiplication.

Another idea: The class chooses an ordered set of positive integers – Each pair writes them down in that order with negative signs in front of any number of the integers. The inner ring is identical, but can be rotated. The winner is the group that finds the minimum sum.

Another Idea: Students do the same as on the previous slides except they swap rings and try to find the highest score to hand back to the owner. However – the rotations MUST move the zeros so they are not coupled. If both players play optimally I think these are the scores for different sizes of rings…  2, 6, 4, 9, 10, 28…