This spinning centrifuge has a matching set of -1 (South) and +1 (North) magnets on the inner and outer ring. The inner ring is the one that spins. You must try to find a combination of +1s and -1s so that when it is rotated away from this all-repulsive position there is an equal number of repulsive and attractive matches.

In this 13-centrifuge we have thirteen magnets in each ring so of course we cannot have an equal number of repulsive and attractive couples. For odd number centrifuges we require that the number of attractive and repulsive couples are within +/-1 of each other. For this rotation we have:

  • 6 attractive
  • 7 repulsive

so this works! However we must check all rotations…

For this rotation we have:

  • 4 attractive
  • 9 repulsive

The attractive and repulsive forces must be within 1 of each other, so this fails. This 13-centrifuge pattern does not work. I’m unsure if any 13-centrifuge pattern works. However, I did find many patterns which did work…

This 15-centrifuge pattern works. No matter how you rotate the inner and outer rings – the number of repulsive couples is always 7. The number of attractive forces is always 8. Isn’t this beautiful! Look through the following slides, but don’t show this success to your students. They should start their exploration with much smaller solutions starting with 2-centrifuges (impossible) and 3-centrifuges (possible). Even if you do choose to show them the 13-centrifuge failure, the correct mathematical response to exploring the centrifuges is to understand what is going on with smaller centrifuges.

7 repulsive couples shown…

7 repulsive couples shown…

7 repulsive couples shown…

7 repulsive couples shown…

7 repulsive couples shown…

7 repulsive couples shown…

7 repulsive couples shown…

Less interesting for mathematicians, but more interesting curricularly is to put numbers 0-n in a ring and then assign + and – to each. The winning ring is the one that – when rotated – keeps the sum of the products of the couples as close as possible.

I’ll make this idea into a mini-competition: “Integral Centrifuge II”

Integral Centrifuge

(MathPickle, 2016)

This puzzle is for students learning about the multiplication of integers. It deals only with the integers +1 and -1, so things are kept focussed on ridiculously easy multiplication with the backdrop of an unsolved problem. The slide above shows a successful and unsuccessful attempt for large centrifuges. Students should start small! They should start by discovering centrifuges like the one pictured below.

Mathematicians may recognize that this puzzle is inspired by Barker codes. Unlike linear Barker codes, here I’m playing with rings.

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)