Filthy rich

Calvin draws 100 lines in the plane such that no two are parallel.

For each equilateral triangle formed by 3 of the lines, he receives $3. For each non-equilateral isosceles triangle, he receives $1.

What is the maximum amount of money Calvin can profit from this venture? (Answer in dollars)


The answer is 4900.

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1 solution

Sharky Kesa
Feb 11, 2017

I will give a generalised proof for n 3 n \geq 3 lines.

Define a based isosceles triangle to be a triangle formed by three of the lines, with two sides of equal length marked. Then the amount of money obtained by Calvin is equal to the number of based isosceles triangles.

For each ordered pair of distinct lines ( l 1 , l 2 ) (l_1, l_2) , there is at most one isosceles triangle with l 1 l_1 unmarked and l 2 l_2 marked. So there are at most n 1 2 \lfloor \frac{n-1}{2} \rfloor based isosceles triangles with l 1 l_1 unmarked. Therefore, there are at most n × n 1 2 n \times \lfloor \frac{n-1}{2} \rfloor based isosceles triangles altogether.

If n n is odd, Calvin can achieve this upper bound by drawing the lines to form the sides of a regular polygon with n n sides. If n n is even, Calvin can achieve this upper bound by drawing the lines to form all but one of the sides of a regular polygon with n + 1 n + 1 sides.

Substituting n = 100 n=100 , we get Calvin profits $4900.

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