Skinny Man Tango starts with a shape and then changes it to another shape based on three simple rules. The results can be as beautiful as watching tango.

To explain the rules I’ll start with this shape.

Replacement Rule: Each edge of your shape must be replaced by two edges of equal length meeting at a right angle. This looks successfully done here, but there is another rule…

No Touching Rule: The new object’s perimeter cannot touch itself. So this does not work.

The new object’s black perimeter touches itself – so this also does not work because of the no-touching rule.

This satisfies both the Replacement Rule and the No-Touching Rule. Unfortunately there the last rule is violated here. ;-(

Minimal Area Rule: The area of the resulting object must be as small as possible.

The area of this object is 236. We can get a smaller area…

The area of this object is as small as possible. This is the next object we seek. Sometimes there will be more than one option. Sometimes there may be none.

Area 153 is as small as you can get.

Let’s start with a much simpler shape and see how many times we can construct a new object…

Our object is going to be this right triangle of height 8 and area 32. Students can copy this on graph paper. Discuss what the next object will look like. Remember the Replacement Rule, No-Touching Rule, and Minimal Area Rule…

This is one option.

Calculate the area. Is it the only solution?

It is not the only option, but this other option is just a mirror image of the first – not very interesting – so I’m just going to explore the first object.

It has area 64. Now find the shape that results if you repeat the algorithm. Remember the rules… Replacement Rule, No-Touching Rule, Minimal Area Rule.

It has area 64.

Does this shape work? What is its area?

Yes – the shape does work, but it is not unique. Find the other possibilities…

A second possibility…

A third possibility…

That’s all. Three shapes. This is the first shape again.

It has an area of 96. What an interesting pattern! 32, 64, 96… What do you expect next?

How many possible shapes can you end up with the next iteration? Here is one.

But there are others…

What is the area?

128. It seems like our pattern is continuing. Is this just luck? 32; 64; 96; 128…

Your class can try to explore this pattern. It can also see how many times a new shape can be found satisfying the three rules. Does this go on forever or does it terminate?

A small fraction of the shapes that can be discovered. The tree does not keep growing. All branches terminate!

Skinny Man Tango

(MathPickle, 2016)

Skinny Man Tango highlights a new and beautiful algorithm that generates a hierarchy of shapes according to a set of rules. It is great for a classroom after they have done basic area calculations. Older students learning square root of two will be able to calculate the perimeter of the shape after each step. It’s a geometric progression. The area seems to be following an arithmetic progression, but can that really continue or is it just luck?

Students should work on normal graph paper… copying shapes from sheet to sheet.

Download a pdf 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)