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. Also, you must place the 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 and I worked out it was a triangle.
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:
Hi to 'The Nrich Team'
My 'top' S4 set were this month inspired by your various 'Rolling Polygon' problems.
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 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).
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!
We do agree! Many thanks.