Solving mazes and Mobius paper chains at the Curious Minds Club (St Thomas of Canterbury Primary School, 13 December 2019)

In our final week of this term at the Curious Minds Club we explored how to solve a maze. A maze is a type of topological puzzle. I showed the children a method of solving a maze by shading in all the dead ends: a dead end is chosen and shaded back to a point where a decision has to be made; at this point no further shading can be done so another dead end is chosen and the shading repeated. When no further dead ends can be found what remains is the solution to the maze. The children explored this method using some Christmas mazes I found online.

Last week we explored Borromean and Brunnian links. There was one activity we ran out of time for, so I brought it this week. Professor Tadashi Tokieda calls this a Borromean ribbon. His Numberphile You Tube video gives a great visual demonstration.

borromean ribbon

I tied some strips of ribbon into closed loops and showed the children how to set them up on one hand. The children soon got to grips with how to pull one ribbon down and come up with a surprising result.

The final activity was a make a Mobius paper chain to take home. I told the children we could have made regular Christmas paper chains, but they are boring. So much better to remember the session we did a few weeks ago on the Mobius loop. I had cut strips out of Christmas themed paper and reminded the children of the twist which produces the Mobius loop. The children then got to work, threading, twisting and taping. Here are some examples:

Year 5 boy:

Y5boyMobiusXmas

Year 6 boy:

Y6boyMobiusXmas

Borromean rings and Brunnian links at the Curious Minds Club (St Thomas of Canterbury Primary School, 6 December 2019)

This week at the Curious Minds Club we explored Borromean and Brunnian links.  We started with the Borromean rings as they are the simplest Brunnian link. I showed the children how to turn coloured pipe cleaners into circles. They made a red one and a white one, then placed the white on top of the red one and threaded a blue one through, in an over-under-over-under sequence. I showed the children that as a ring of three this is a strong link, but if you remove any one ring the whole thing falls apart.

200px-BorromeanRings.svg

I asked the children to turn their flat ring into a three-dimensional structure, almost like a gryoscope.

220px-3d_borromean_rings_by_ronbennett2001

A drew a table to show that each ring is outside one ring and inside another ring.

Outside Inside
White Red Blue
Blue White Red
Red Blue White

I surprised the children by asking them how their Borromean ring is like the game ‘Rock, Paper, Scissors’. One girl was able to identify that each component has a different relationship with the other two components. I drew the table below and pointed out that white occupies the same positions in the table as rock; that blue corresponds to paper; and that red corresponds to scissors. I explained that the concept of ‘beating’ equates to being ‘outside’ and the concept of ‘losing to’ equates to being ‘inside’.

Beats Loses to
Rock Scissors Paper
Paper Rock Scissors
Scissors Paper Rock

By this point some of the children had discovered that they could turn their Borromean ring inside out. I encouraged all the children to practice turning their ring inside out and back again. I think this is one of the most fascinating aspects of the Borromean ring and is due to the fact that no two of the three rings are actually linked with each other.

We then moved on to making a Borromean ring of a Borromean ring! Yes, we went fractal! I asked the children to take three pipe cleaners of one colour, three of a second colour and three of a third colour. They made normal Borromean rings with their first and second colours and placed one on top of the other. They made another Borromean ring with their third colour to get the twists in the right place, undid the ends, threaded it through and retied the ends. When made three dimensional this is a very interesting object to look at.

Some children had raced a bit ahead, so I asked them to make triangles instead of circles and thread them into a ring. I showed them one I had made earlier. Two children thought the circles made a better three dimensional shape, but one boy argued in favour of his triangles. He had indeed succeeded in making something harmonious to look at.

The final task was a make a 4 component Brunnian link, with four different coloured pipe cleaners. This is trickier than the Borromean rings as the green link has to be made much smaller, but the threading is not too complicated. With a little assistance the children were able to complete theirs. I asked what would happen if one link was removed. Due to the effort involved in making it some of the children were reluctant to try this on theirs! I let them use my demo instead, and they were surprised to see the whole thing fall apart. I explained that as this happens with the Borromean rings we should expect it with the Brunnian link as they are part of the same class of links.

200px-Brunnian-4loops.svg

Here are some photos of the children’s work.

Year 2 girl:

 

 

Year 4 girl:

Y4girl Borromean

Year 5 boy:

Y5boy Borromean

Year 6 boy:

Y6boy Borromean

Year 6 girl:

Y6girl Borromean

Perplexing paperclips at the Curious Minds Club (St Thomas of Canterbury Primary School, 18 October 2019)

We did some perplexing things with paperclips and rubber bands this week. I showed the children how to bend a long strip of paper into an ‘s’ shape, then squash it together and add two paperclips at the two places where the paper touches. When I pulled on each end of the paper the children were surprised to see the two paperclips land on the table linked together.

The children then followed a set of drawings I had made, each one getting progressively harder and involving adding in one rubber band, then two rubber bands, then changing the positions of the bands along the strip of paper. The results were a combination of the paperclips and rubber bands being linked together in lots of different ways, some falling off the paper and some staying on. Some children made it to drawing 10, which results in an amazing Borromean link!

If you want to see how the paperclips become linked together, here is a slow motion video:

This is what the table looked like at the end! Lots of mess and lots of fun.

Table after Perplexing Paperclips 18 October 2019