# Problem of the Month(August 2013)

What is the smallest square that contains circles of radius 1 through n so that no circle's center is located in the interior of any other circle? In other words, if d(x,y) denotes the distance between the planar points x and y, and if for 1≤i≤n the center of the circle of radius i is ai, then we want d(ai,aj) ≥ max{i,j}. The cases for small n are easy, but get difficult quickly.

What if the centers need to be contained in the square, but the rest of the circle need not be? Again we ask for the smallest square that works.

When the entire circles need to be contained in the square of side s, the smallest-known sides are shown below:

 n=1s = 2 n=2s = 3 + √2 = 4.4142+ n=3s = 5 + 3/√2 = 7.1213+ n=4s = 7 + 2 √2 = 9.8284+ n=5s = 12.5604+ n=6s = 15.4695+ n=7s = 18.4994+ n=8s = 21.6882+ (Joe DeVincentis) n=9s = 25.3951+ (Maurizio Morandi) n=10s = 28.6835+ (Joe DeVincentis) n=11s = 32.5383+ (Joe DeVincentis) n=12s = 36.4636+ (Joe DeVincentis) n=13s = 40.3575+ (Joe DeVincentis) n=14s = 44.2073+ (Joe DeVincentis) n=15s = 48.2039+ (Joe DeVincentis)

When only the centers of the circles need to be contained in the square of side s, the smallest-known sides are shown below:

 n=2s = √2 = 1.4142+ n=3s = 2 + 1/√2 = 2.7070+ n=4s = 3.8724+ n=5s = 4.9836+ n=6s = 6.6691+ n=7s = 8.7329+ (Joe DeVincentis) n=8s = 10.7169+ (Joe DeVincentis) n=9s = 13.0749+ (Joe DeVincentis) n=10s = 15.4216+ (Joe DeVincentis) n=11s = 17.9409+ (Joe DeVincentis) n=12s = 20.2186+ (Joe DeVincentis) n=13s = 22.7292+ (Joe DeVincentis) n=14s = 24.9614+ (Joe DeVincentis) n=15s = 27.6147+ (Joe DeVincentis)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/1/13.