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Speeding Up, Slowing Down

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.

How Far Does it Move?

Stage: 3 Challenge Level: Challenge Level:1

This problem clearly got a lot of you thinking! Several of you sent in the correct answer, including Gemma, Rachel, David, Alex, Charlie, Robert, Joel, Jamie, Carys, Soph, Bex, Rhi, Joe, Veronica, Harriet and Elspeth, all from Cowbridge Comprehensive School.

As Azeem of Mason Middle School states:

You have to have a triangle with 40 radius. Also, you must place the red dot at the bottom left corner of the triangle.

Some of you worked it out using some systematic thought and trial and error.

Sathya and Michael made a good effort at explaining how they worked this out .

Sathya from Scots College, New Zealand considered whether or not the dot could be placed in the centre of the shapes:

I first checked whether it could have been the centre of the shapes. This is impossible as it always results in a linear graph.

Sathya then went on to consider how many sides the polygon might have:

Then I checked how many segments in the line (changes in the shape of the line) and found there was 1 for every side of the shape i.e. for a pentagon 5, square 4 etc. Therefore, the graph shown had 3 segments, so I worked out it was a triangle (obviously!).

Michael from St John Payne School went a little further in exploring where the dot might be placed:

First I knew it had to be on the vertex because there was a part of the graph that was flat.
The only point at which the dot isn't travelling anywhere when the polygon is rolling is on a vertex, because then the dot is always in contact with the floor.

He then looked at the units on the graph to consider which vertex the point would be on:

This told me the time period for one turn about a vertex was roughly 2.5 units as this was the length of time for which the gradient was zero.
This meant that the polygon had gone through 2 cycles before it was going to pivot on the vertex that held the dot (because there was roughly 5 units of time before the gradient was zero and 5/2.5 = 2) .
This meant that the red dot had to be 2 vertices after the bottom right vertex.
This gave enough grounds for trial and error to find the right polygon fairly quickly.

Thank you to Ken Nisbet, the class teacher of 4YP at Madras College in St Andrews, Scotland, for sending us this message and the very impressive work carried out by some of his students:

Hi to 'The Nrich Team'

My 'top' S4 set were this month inspired by your various 'Rolling Polygon' problems.
We decided to investigate the total distance travelled by the dot, when it was placed at one of the vertices, from 'take-off' to 'landing' when each polygon had side length 1 unit.

The class divided into groups of 2 or 3 students. Exact values were the order of the day. The Equilateral triangle and Square were fine though much discussion was needed regarding the final form of the answers. The pentagon proved a lot more challenging with the Golden ratio eventually surfacing.
The overall pattern in the answers were quite remarkable with a very curious sequence emerging:
1, 2, 5, 12, ...

The groups spent over a week working on this investigation with excitement mounting as the sequence developed. Predictions were made at the stage when the hexagon revealed the number 12 as the 4th term. For the Septagon exact values were not possible so conjectures for the 5th term of the sequence were tested using very accurate calculator work (Sine Rule & Cosine Rule etc).
What was a complete surprise was that a non-integral value for the 5th term emerged. There are not many situations, at this level of work, where approximate values can be harnessed to disprove an integer sequence conjecture. "This must be wrong" was the general feeling and all the sums were checked again and again... but eventually calculations by three separate independent groups confirmed the result.

I feel that the challenge and sheer range of technique required for this investigation has benefited my students immensely. They are aged 14 to 15 and produced work of impressive depth and quality. Thanks for the stimulation that your questions have provided ... keep up the great work. I have attached the write-up produced by David, Nicholas and Robert as it was a superb exposition.... I'm sure you will agree!
All the best Ken Nisbet

We do agree! Many thanks.