Imagine a spider going for a morning walk around the web. At each intersection he rolls a dice (carefully so it doesn’t stick) to determine the next direction. If the dice point him towards an intersection already visited – he rolls again. How likely is he to visit all the intersections on his web?

We can choose any web, but let’s restrict ourselves. Let’s say that the web has to lie flat. We want beautiful webs that we can be proud of – not the black widow 3D mass of chaotic silk. Our spider is proud to be spinning all his webs in 2D.

Our spider starts anywhere he wishes.


What fraction of success can our spider achieve?

In this case we can see that he has failed.


It is possible for our spider to succeed, but this is pretty unlikely. In this spider grid our spider only wins a small fraction of the time.

What are the largest fractions possible?


This spider web looks a bit better. Calculate the odds of winning starting from any intersection you choose…


The answer is either 1/8 or 1/16.

Go away and find the highest few fractions possible.

Spoiler alert – stop here and experiment!


This graph yields some of the highest rates of success. Find the fraction of wins starting from each intersection.

Of course 1 is the highest fraction attainable. If our spider starts from this point he is guaranteed a win. 5/6 is still pretty good! It is, however, not the second best fraction possible. Look for the second best fraction.

Spoiler alert – stop here and experiment!

I do not know that this octahedral net is the one that produces the second highest fraction, but I think it is. Find the fraction of success starting from any intersection.

All intersections on the octahedral net are equivalent. The chance of success is a whopping 8/9!

Compete to find the top ten fractions- starting with 1.

I do not know the answer to this problem.


These are the top 5 that I’ve found, but I probably have overlooked something.

$100 reward for the person who finds, proves and publishes the top ten fractions.


Spider Walk

(MathPickle, 2016)

Educators: Explore probability by randomly wandering around a spider web. At each intersection randomly choose a neighbour that has not been visited. What is your probability of success?

Mathematicians: What fractions are possible? What web (a planar graph) gives the highest probability of success if success is not guaranteed?

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)