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Up and Across

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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 red 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 red dot should be placed:

Then I knew that as the highest points were at the start, the red dot must go on the centre of the side that is 1 anti-clockwise of the top side on a hexagon with a diameter of 45!

Robert from Cowbridge Comprehensive sent in a solution with slightly different reasoning:

I already knew that the radius was 45. The height graph had six distinct curves before it repeated, so the shape had to have six sides.

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, of radius 45, with the dot in the middle of the upper left side.

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.