Problem of the Month
(June 2016)

This month we consider the Math Magic problem of March 2013, but for equilateral triangles instead of circles. Consider an infinite grid of unit squares. We want to place k equilateral triangles on this grid so that: 1) no two triangles cover part of the same grid square and 2) the set of squares that are at least partially covered by a triangle form an m×n rectangle. For a given m and n, what is the minimum k that will work? Below are some examples.

ANSWERS

Answers were submitted by Joe DeVincentis.

Let f(m,n) be the smallest k that works for a given m and n.

It is clear that f(1,n) = f(2,n) = n/3 and f(a+b,n) ≤ f(a,n) + f(b,n).

Joe DeVincentis showed the following general bounds for large n:


f(3,7n+3)≤2n+1


f(4,4n)≤n


f(5,9n–7)≤2n


f(5,9n–2)≤2n+1


f(6,5n–7)≤n


f(6,11n–3)≤2n+1


f(6,11n–9)≤2n

Smallest-Known Number of Equilateral
Triangles Covering m×n Rectangle
m\n1234567891011121314151617181920
1 11122 23334 44555 66677
2 11122 23334 44555 66677
3 1 1 EF 2 2 2 3 3 3 EF 4 4 4 5 5 5 JD 6 6 6
4 2 2 2 2 2 2 2 JD 3 3 3 EF 4 4 4 EF 5 5 5 JD
5 22223 33444 44555 6 6 6 6 JD
6 2 2 2 2 3 3 EF 4 4 4 4 4 JD 5 5 5 5 JD 6 6
7 3 3 3 2 3 EF 4 4 4 JD JD JD 4 JD JD JD JD JD JD JD
8 3 3 3 JD 4 4 4 JD JD 5 JD JD JD JD JD JD JD JD JD 7
9 3 3 3 3 4 4 4 JD 5 5 JD JD JD JD JD JD JD JD JD 7
10 4 4 EF 3 4 4 JD 5 5 5 5 JD JD JD JD JD JD JD JD 7
11 4 4 4 3 4 4 JD JD JD 5 JD JD JD JD JD JD JD JD JD JD
12 4 4 4 EF 4 4 JD JD JD JD JD JD JD JD JD JD JD JD JD JD
13 5 5 4 4 5 JD 4 JD JD JD JD JD 8 JD JD JD JD JD JD JD
14 5 5 5 4 5 5 JD JD JD JD JD JD JD
15 5 5 5 4 5 5 JD JD JD JD JD JD JD
16 6 6 5 EF 6 5 JD JD JD JD JD JD JD
17 6 6 JD 5 6 5 JD JD JD JD JD JD JD
18 6 6 6 5 6 JD JD JD JD JD JD JD JD
19 7 7 6 5 6 6 JD JD JD JD JD JD JD
20 7 7 6 JD JD 6 JD 7 7 7 JD JD JD

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 6/1/16.