# Problem of the Month (March 2012)

Given two shapes X and Y, what is the largest connected shape S so that you can fit a total of n non-overlapping copies of S inside X and Y? We got interested in this problem for X = unit equilateral triangle and Y = unit square, but both X = unit square and Y = circle of diameter 1, and X = unit equilateral triangle and Y = circle of diameter 1 are bound to be interesting as well. Even X = square of side 2 and Y = square of side 3 has some non-trivial behavior.

Maurizo Morandi, Berend van der Zwaag, Joe DeVincentis, Richard Sabey, Bryce Herdt, Andrew Bayly, and Jeremy Galvagni sent solutions.

Jeremy Galvagni got me to bold the entries that are known to be optimal.

Largest Known Solutions for Triangle and Square
 n=1area = 1 n=2area = 1 n=3area = 1.195+ (MM) n=4area = 20√2–27 = 1.284+ (MM) n=5area = 5/4 = 1.25 n=6area = 3√3/4 = 1.299+ n=7area = 1.332+ (MM) n=8area = 4/3 = 1.333+ (BZ) n=9area = 3√3/4 = 1.299+ (MM) n=10area = 1.359+ (MM) n=11area = 11/8 = 1.375 (MM) n=12area = (24–7√3)/9 = 1.319+ (MM) n=13area = 1.379+ (MM) n=14area = 7/5 = 1.4 (MM) n=15area = 1.340+ (MM) n=16area = (63–34√3)/3 = 1.370+ (MM)

Largest Known Solutions for Circle and Triangle
 n=1area = π/4 = .785+ n=2area = .844+ (JD) n=3area = (3√3 + 4π)/16 = 1.110+ n=4area = 1.029+ (MM) n=5area = 5π/16 = .981+ n=6area = (3√3 + 4π)/16 = 1.110+ n=7area = 7√3/12 = 1.013+ n=8area = 1.100+ (MM) n=9area = 3π/8 = 1.178+ n=10area = 5π/14 = 1.121+ (MM) n=11area = 11π/32 = 1.079+ (MM) n=12area = 1.150+ (MM) n=13area = 1.131+ (AB) n=14area = 7π/20 = 1.099+ (AB) n=15area = 1.159+ (BZ) n=16area = 4π/11 = 1.142+ (MM)

Largest Known Solutions for Square and Circle
 n=1area = 1 n=2area = π/2 = 1.570+ n=3area = (3√3 + 2π)/8 = 1.434+ n=4area = π/2 = 1.570+ n=5area = 1.596+ (MM) n=6area = π/2 = 1.570+ n=7area = 1.618+ (MM) n=8area = π/2 = 1.570+ n=9area = 1.664+ (JD) n=10area = 1.637+ (JD) n=11area = 55/36 = 1.527+ (MM) n=12area = π/2 = 1.570+ n=13area = 1.537+ (JD) n=14area = 1.598+ (JD) n=15area = 1.650+ (BZ) n=16area = 8/5 = 1.6 (BZ)

Largest Known Solutions for Squares of Sides 2 and 3
 n=1area = 9 n=2area = 9 n=3area = 21/2 = 10.5 n=4area = 11.293+ (MM) n=5area = 45/4 = 11.25 n=6area = 12 n=7area = 49/4 = 12.25 (BZ) n=8area = 12 n=9area = 11.823 (JD) n=10area = 12.536+ (JD) n=11area = 99/8 = 12.375 (MM) n=12area = 12 n=13area = 13 n=14area = 63/5 = 12.6 (MM) n=15area = 12.093+ (BZ) n=16area = 112/9 = 12.444+ (MM)

Largest Known Solutions for Triangles of Sides 2 and 3
 n=1area = 9√3/4 = 3.897+ n=2area = 9√3/4 = 3.897+ n=3area = 21√3/8 = 4.546+ n=4area = 3√3 = 5.196+ n=5area = 45√3/16 = 4.871+ n=6area = 3√3 = 5.196+ n=7area = 21(63–34√3)/16 = 5.394+ (MM) n=8area = 3√3 = 5.196+ (MM) n=9area = 3√3 = 5.196+ (MM) n=10area = 20(2–√3) = 5.358+ (MM) n=11area = 11(45–17√3)/32 = 5.347+ (MM) n=12area = 3√3 = 5.196+ n=13area = 13√3/4 = 5.629+ n=14area = 5.417+ (MM) n=15area = 5.155+ (MM) n=16area = 16(9–5√3) = 5.435+ (MM)

Largest Known Solutions for Circles with Diameter 2 and 3
 n=1area = 9π/4 = 7.068+ n=2area = 9π/4 = 7.068+ n=3area = 7.809+ (JG) n=4area = 8.666+ (JG) n=5area = 8.796+ (JG) n=6area = 3π = 9.424+ (JG) n=7area = 9.568+ (MM) n=8area = 9.373+ (MM) n=9area = 3π = 9.424+ (AB) n=10area = 9.658+ (JD) n=11area = 9.600+ (JD) n=12area = 3π = 9.424+ (JD) n=13area = 9.759+ (JG) n=14area = 9.768+ (JG) n=15area = 9.606+ (JD) n=16area = 9.736+ (JD)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/21/12.