Figure of Eight

Well done Andrei from Tudor Vianu National College, Bucharest, and Michael from Portsmouth for sorting this one out. 

There are nine gaps and the top and bottom lengths on the "Figure of Eight" must be parallel.

This means that the two sides, from top down to the bottom, must each cover the same number of pegs, because of the symmetry.

Those sides could be $1$, $2$, or $3$, but not $4$ because that only leaves one gap ($9-2\times4$) for top and bottom between them!

Working through those possibilities systematically:

$1$ for the sides leaves $7$ ($9 - 2$) for the top and bottom: $1$ with $6$, $2$ with $5$, or $3$ with $4$.

$2$ for the sides leaves $5$ ($9 - 4$) for the top and bottom : $1$ with $4$, or $2$ with $3$

$3$ for the sides leaves $3$ ($9 - 6$) for the top and bottom : so just $1$ with $2$

Between two adjacent pegs the angle at the centre is $40^\circ$

A chord between any two pegs and the centre of the pegboard makes an isosceles triangle.

Split that triangle along its line of symmetry to get a right-angled triangle and then use trigonometry to see that :

when the pegs are $n$ gaps apart, and the radius of the board is taken as $1$, the length of the chord is $2\sin{20n^\circ}$ 

 

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Click here for a copy of the Excel file Figure of Eight Results