Figure of Eight
On a nine-point pegboard a band is stretched over 4 pegs in a
"figure of 8" arrangement. How many different "figure of 8"
arrangements can be made ?
Age
14 to 16
Challenge level
Problem
On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement so that one pair of opposite sides cross and the other pair are parallel.
Image
How many different "figure of 8" arrangements can be made ?
Which of these arrangements has the area of the lower (green) portion nearest to double the area of the upper (yellow) portion ?
For printable sets of circle templates for use with this activity, please see Printable Resources page.
Getting Started
Identify why the yellow and green triangles are similar and then try to work out the line ratio.
You'll need to think about the angles made to the centre of the pegboard.
How is the area ratio connected to the line ratio?
Student Solutions
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 $2\sin{20n^\circ}$
Image
AB is parallel to CD which means that both the yellow and
green triangles have the same set of angles.
So, because the yellow and green triangles are similar, the
ratio of their areas is the square of the ratio for lengths.
And, from the paragraph above, we know that the side lengths
can always be calculated.
$\frac{AREA CDE}{AREA ABE}$ near $2$ implies that
($\frac{CD}{AB})^2$ must be near 2
Image
Along the top are the number of gaps on the pegboard that $AB$
spans.
Down the side are the same for $CD$
Beneath or beside each of those are the lengths calculated
using $2\sin{20n^\circ}$
The table shows the ($\frac{CD}{AB})^2$ values.
The combinations that do not make a Figure of Eight with $AB$
and $CD$ parallel are blanked out.
There are two values near $2$ but $1.815$ is the nearer so the
solution is when yellow has two pegs and green has three pegs
Click here for a copy of the Excel file Figure of Eight Results
Teachers' Resources
This is a demanding problem because it requires several sub-problems to be solved to make a successful sequence of reasoning culminating in the final result.