I've Got The World On A String
Madeleine wishes to create the design above using a wooden board, nine nails, and a length of string. She will hammer the nails into the board ( almost all the way), arranged as the vertices of a regular nonagon. She will then take the string, tie one end to one of the nails, then wrap the string around the nails repeatedly until every pair of nails has exactly one length of string connecting them.
Madeleine plans to use a piece of string 10 meters long. What is the farthest apart that she could place the nails at consecutive vertices of the nonagon?
Give your answer in millimeters, rounded to the nearest whole millimeter.
Details and Assumptions:
- The nails have negligible width, and none of the string is used up when it is wound around a nail.
The answer is 134.
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1 solution
Let the nails lie on a unit circle on complex plane with the first one on (1,0). The nails will therefore represent the 9 roots of unity. There are total four different types of lengths to be measured, each repeated 9 times. If the first four roots of unity are a, b, c, and d, then the corresponding lengths are represented as |a-1|, |b-1|, |c-1|, and |d-1| respectively. And each of them being equal to 2sin40, 2sin80, 2sin120, and 2sin160 (magnitutes of complex numbers).. Hence, total length of string required is 9x2x(sum of sines). And side length is equal to 2sin40. Using a little trignometry, the ratio comes out to be around 74.62. Dividing 10000 mm by 74.62 we get 134 mm as the maximum side length.