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.