The Three Utilities Problem at the Curious Minds Club (St Thomas of Canterbury Primary School, 11 October 2019)

An investigation of that classic problem in topology, the Three Utilities Problem. The children made their first attempt in flat, two-dimensional space i.e. on a mini whiteboard, so they could rub out their attempts and keep trying. After a few minutes I explained that the problem cannot be solved in two-dimensional space as there is not enough space! I asked the children what we could do in this case, and one boy was very quick to suggest trying it in three-dimensional space.

I produced four big tori, in the form of an inflated swim ring which I had wrapped in white duck tape and labelled, so that the dry wipe markers could be used and rubbed off as needed. After a few attempts, and some hints from me that they needed to use the whole length and circumference of the torus, we had some correct solutions.

three utilities

I then reminded the children that a torus is topologically equivalent to a coffee mug, and that if they could solve this problem on a torus they could also solve it on a mug. I then produced four white mugs which I had labelled. The dry wipe markers rub off very easily on a mug. The children spent the rest of the session exploring how to solve the problem. They needed a few hints about using the handle and the base, and going under the handle, but most of them got there in the end. The children seemed to really enjoying solving the problem using such unusual materials.

three utilities mug

Deforming a doughnut into a coffee mug at the Curious Minds Club (St Thomas of Canterbury Primary School, 20 September 2019)

A continuation of our exploration of topology. An explanation of the rules of deformation (no new holes, no filling in holes, no gluing, no tearing), followed by getting out the modelling clay and deforming a torus (or doughnut) into a coffee mug. An extension of this by deforming a two-hole torus into a pair of pants, and a three-hole torus into a vest (after explaining why a vest has three holes).

A new activity to demonstrate that topology includes changing an object’s shape without changing its size. Cutting a hole the size of a 5p in a piece of paper, trying to pass a 2p through this hole, finding this is impossible in 2D space, switching to 3D space by lifting up the piece of paper and introducing height, and finding that the 2p will pass through the hole once the paper is twisted and manipulated.