Richard Smalley was awarded the Nobel prize in Chemistry for the discovery that he could make a very very small soccer ball out of Carbon atoms. In 2005 he suddenly began to shrink.

He ended up so small that he could bounce on the little trampolines that make up graphene. (Graphene is a single layer of carbon atoms in this hexagonal pattern. The carbon atoms are at all the intersections where three connectors come together.)

Richard Smalley spent his time making graphene-trampoline puzzles.


Here is one of them…

Richard Smalley takes a dice out of his pocket and starts rolling it. Can you guess what number he needs to get before he can move on?


He needs to keep rolling till he gets a “1.” He got it and moved in that direction. Now he needs to roll…

… another “1.”

What number must he roll now?

He must keep on rolling till he rolls a 1 or 2.

He rolled a “2.”

Now what number or numbers must be rolled?

A “2” or “6.”

He rolled a “6.”

This path is a failure because not all graphene trampolines were visited. Are all paths starting from the lower left trampoline failures?

This path is a success because all graphene trampolines were visited. Because a path failed and a path succeeded we color this trampoline yellow.

To solve a graphene-trampoline puzzle you must color each trampoline:

Green – all paths succeed.

Yellow – some paths succeed – some paths fail.

Red – all paths fail.

What color should this trampoline be?

Here is a success… so it cannot be red.


Here is another success.

These are the only two ways that dice rolling could take you… so this trampoline needs to be colored green (all paths succeed.)


Now we must color all the other trampolines.


What color does this trampoline get?

Do paths starting here always succeed? Sometimes succeed? Always fail?



This looks good…


Except we missed two trampolines… so this is a failure.

This also fails.

…and this one fails too. It seems there are no successes from this starting trampoline so we color it red.

Let’s find the color of another trampoline…

Let’s choose this one. Everyone take 30 seconds quietly to figure out what color should be used.

Does everything fail?

…or does some path succeed?

It looks bad…

It’s red – right?

Wrong – here is a path that works. So we need to use yellow (sometimes fails – sometimes succeeds.)

So we know what to color three of the trampolines. Take 20 seconds to figure out another trampoline’s color.

This is the solution to the puzzle.

Students in higher grades can figure out the probability of success from each yellow trampoline.

Then it is time to pair up with a green, yellow and red writing utensil. Spoiler alert – do not go on to the next page unless you want the answer to this puzzle sheet.

Download puzzle sheets like this one and harder here. This includes puzzle sheets that are good for giving older students practice with probability.

For young students, you may wish more options for puzzle sheets. Here is the alphabet in Graphene Trampoline font.

As a really tough problem (which I am only part way to solving) I’d like to know the largest fractions possible for these Graphene Trampoline puzzles. The probability of success for this Trampoline in this puzzle is one of the highest. Try to calculate it.

The largest fractions I’ve discovered are 1, 5/6, 29/36, 3/4. I will give $100 to anybody who can find, prove and publish the complete list of possible fractions greater than 1/2.

I also have a more general version of this question which asks the same for any planar graph.

Graphene Trampoline

(Gordon Hamilton, 2005)

This puzzle has broad applicability. Younger students will enjoy finding the paths. Older students will be challenged to find the probability of success. Download puzzle sheets 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 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

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)