# What is the smallest rectangle that will contain all squares of size 1 to 33?

At Tightly Packed Squares, best solutions for square packings for 1 to 32 squares are given.

What would the best packing for 33 squares be? Here are various rectangles that might work, preceeded by the excess squares.

0 -- $67\times187$, 1 -- $70\times179$, 4 -- $83\times151$, 6 -- $109\times115$, 11 -- $110\times114$, 14 -- $111\times113$, 15 -- $112\times112$, 16 -- $65\times193$, 17 -- $102\times123$, 20 -- $89\times141$, 22 -- $77\times163$, 26 -- $93\times135$, 27 -- $86\times146$, 29 -- $91\times138$, 31 -- $80\times157$, 32 -- $79\times159$, 37 -- $103\times122$, 38 -- $71\times177$, 44 -- $99\times127$, 47 -- $96\times131$, 51 -- $85\times148$, 55 -- $104\times121$.

For 34 squares, possible best rectangles are

0 -- $115\times119$, 3 -- $116\times118$, 4 -- $117\times117$, 5 -- $74\times185$, 7 -- $84\times163$, 9 -- $82\times167$, 10 -- $83\times165$, 11 -- $107\times128$, 14 -- $103\times133$, 15 -- $100\times137$, 18 -- $71\times193$, 21 -- $89\times154$, 23 -- $92\times149$, 31 -- $108\times127$, 35 -- $98\times140$, 39 -- $94\times146$, 40 -- $75\times183$, 43 -- $104\times132$, 46 -- $69\times199$, 49 -- $109\times126$, 51 -- $101\times136$, 56 -- $91\times151$, 61 -- $87\times158$, 65 -- $110\times125$, 67 -- $72\times191$

Ironically, human minds seem to be very good at these problems. Most of the higher order square-packing results for 50+ differently sized squares were solved by hand. Can anyone bring a new methodology for solving this problem?

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Are those solutions up to 32 all proved to be best, or just the best found so far? –  Noam D. Elkies Oct 3 '11 at 3:18
In 2004, Shigeyoshi Kamakura found a packing for squares 1-23 in a 66x66 square, which went against a longstanding impossibility proof. Many of these proofs are programmatic, and at least one program is known to have failed. That said, I mostly trust the programs and handwork that has gone over smaller results. –  Ed Pegg Jr Oct 3 '11 at 12:20