Congratulations to those of you who deduced the correct answer to this problem.
Cameron from Boroughbridge Primary was one of the first to send in a solution. He used a combination of observation and trial and error to begin the problem:
If you put the dot in the centre of a side and look at the height chart, it goes down and then immediately up again. This also happens on the height chart that you are trying to make. If you try this with different shapes, a hexagon makes the right pattern.
He then considered on which side the dot should be placed:
Then I knew that as the highest points were at the start, the dot must go on the centre of the side that is 1 anti-clockwise of the top side on a hexagon.
Robert from Cowbridge Comprehensive sent in a solution with slightly different reasoning:
The height did reach zero, but it did not stay at zero, meaning that the dot could not be on a corner, but had to be in the middle of a side.
The height graph had four of the curves before hitting zero, so the dot had to be on the fourth side to touch the ground.
Therefore the shape was a hexagon, with the dot in the middle of the upper left side.
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! All the best Ken Nisbet