Tightly Packed Squares

color scheme

SouthwestColors

number of squares

32

What is the smallest rectangle that can hold all the squares of sizes 1 to

n

? This problem is unsolved for more than 32 squares. The excess area in these packings is 0,1,1,5,5, 8,14,6,15,20, 7,17,17,20,25, 16,9,30,21,20, 33,27,28,28,22, 29,26,35,31,31, 34,35. How the excess is bounded for higher

n

is an unsolved problem, but the bounds seem to be

n/2

and

2n

.

Details

Richard E. Korf, "Optimal Rectangle Packing: New Results," 2004.

Eric Huang and Richard E. Korf, "New Improvements in Optimal Rectangle Packing," 2009.

Ed Pegg Jr, "Square Packing," 2003.

External Links

Square Packing (Wolfram MathWorld)

Permanent Citation

Ed Pegg Jr, Richard E. Korf

"Tightly Packed Squares"
http://demonstrations.wolfram.com/TightlyPackedSquares/
Wolfram Demonstrations Project
Published: January 1, 1999