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Rainbow Squares is a challenge inspired by Henri Picciotto. Each rainbow arc needs to be anchored at two integers that sum to a square. Here we have tried to find a solution for 1-10, but have failed because the last two numbers, 2 and 3, do not sum to a square. Is this possible?

Downloadable Rainbow Square puzzle-sheets here.

1-10 did not work. It was impossible. However, 1-26 does work. Complete the two puzzles on the left.  Spoiler alert: the next page will give the answers.

Solution to the previous slide.

Another pair of puzzles with the solution on the next page...

Solution to the previous slide.

Here I've failed with 1-60 because the remaining numbers, 21 and 22, do not add to a square. Even though there are over 4 million solutions to this it took me more than an hour to find one! Computers are reliably good at these kind of brute force searches. I choose not to tell students how good computers are because everyone wants to feel awesome when they solve something difficult - not be told that a computer found 4 million solutions in one second. It is a matter of our human pride 😉

Rainbow Squares

(Henri Picciotto, 2014)

This was co-winner of the 2016 George Pólya Award by the Mathematical Association of America. Download the original paper here.

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)