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

.