Tightly Packed Squares
What is the smallest rectangle that can hold all the squares of sizes 1 to ? 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 is an unsolved problem, but the bounds seem to be and .
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.