Pair up students. Each student writes 0 to 4 in a ring and then secretly assigns + and – to each. Later on they can challenge each other with 0 to 5 rings or even larger.


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…

Integral Centrifuge II

(MathPickle, 2016)

This group challenge is for students learning about the multiplication of integers. Groups or individuals compete against each other to try to optimally score by rotating integral rings. Here is a pdf of some puzzle sheets.

Mathematicians will be more interested in the first Centrifuge puzzle that uses only 1s and -1s.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, etc.

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)