Problem of the Month (January 1999)

If we have n distinct points in the plane, they determine n(n-1)(n-2)/6 triangles, and therefore 3n(n-1)(n-2)/6 angles. Each of these angles θ satisfies 0 ≤ θ ≤ π, with equality if the points are colinear. How often can an angle θ occur?

Let A(n,θ) be the maximum number of times angle θ can be formed by n points. What are the values of A(4,θ)? How about A(n,θ) for larger n? Can you prove your answers?


Joseph DeVincentis gave answers for n≤7. In most cases, his results are the best known.

John Hoffman gave many bounds for A(n,θ). For example, he found that A(2n+1,θ) ≥ n2 and A(2n,θ) ≥ n(n-1) by putting one point at the intersection of two lines and evenly dividing the other points on the two lines.

Hoffman can show that for large n, A(n,π/4) ≥ (1-ε)n2 . I can show the same thing for θ=π/2, using a square grid of points. Hoffman and I conjecture that if 0<θ<π, then log A(n,θ)/ log n → 2.

Hoffman and Devincentis showed that A(n,0)=2n(n-1)(n-2)/6 and A(n,π)=n(n-1)(n-2)/6. No more than two angles in any triangle can be θ=0 and no more than one angle can be θ=π. Arranging points in a line achieves both these bounds.

Generalizing this, one can show A(n,θ) ≤ A(n-1,θ) n / (n-3). To see this, consider all n collections of n-1 points out of n points. In each collection, at most A(n-1,θ) angles θ occur, and each such angle appears in (n-3) collections.

There are some general constructions which provide lower bounds. A(n,2π/n) ≥ n(n-2) by putting the points at the vertices of an regular n-gon. Similarly, if θ ≤ 2π/(n-1), then A(n,θ) ≥ (n-1)(n-2) because we can put the points at equally spaced points on the circumference of a circle.

Here are the best known bounds for n=3, 4 and 5.

Angles Formed by Three Points

θ A(3,θ)
[ 0 , π/2 ) 2
[ π/2 , π ] 1
π/3 3

Angles Formed by Four Points

θ A(4,θ)
( 0 , π/3 ] 6
( π/3 , π/2 ] 4
( π/2 , 2π/3 ] 3
( 2π/3 , π ) 2
0 8
π/4 8
2π/5 5
π 4

Angles Formed by Five Points

θ A(5,θ)
( 0 , π/4 ) 12 , 15
( π/4 , π/2 ] 8 , 15
( π/2 , π ) 5 , 7
0 20
π/5 15
π/4 16 , 20
π/3 11 , 15
2π/5 10 , 12
π 10

Here are the conjectured values of A(n,θ) for some special angles. Can you verify or continue any of these sequences?

Special Angles

θ \ n 34 567
0 28204070
π/6 26122436
π/4 28162232
π/3 36112736
π/2 1481421
2π/3 135813
π 14102035

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/28/99.