Matched: the best known value
Packing 37 unit squares in a square
What is the smallest square that holds 37 squares of side 1, without overlap? The best known answer is a side of 6.5986196092, from David W. Cantrell, 2002. Velmar Cut matched it.
The numbers
| Best known side | 6.5986196092 |
|---|---|
| Best known side, in full | 6.59861960924436 |
| Side from Velmar Cut | 6.5986196092 |
| Side from Velmar Cut, in full | 6.598619609246030 |
| Difference, Velmar Cut minus best known | +1.670E-12 |
| Found by | David W. Cantrell, 2002 |
| Status | Best known, not proved |
| Time to reach it | 39.892 s, one run |
| Threads used | 16 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 22 upright, 15 turned |
| A plain grid needs | side 7 |
The time to reach this result on one desktop PC.
The problem at 37 squares
Rows need side 7 for one square more than a grid of side 6 holds. As at 11, the best known side has no concise closed form and is a root of a polynomial of degree 8. This layout has twenty-two upright squares and fifteen turned.
A plain grid: side 7
Best known: side 6.5986196092
Blue squares stand upright, orange squares are turned. Both pictures are at the same scale.
History
The best known value at this size is the result of David W. Cantrell, 2002, and it is their result that Velmar Cut equals.
Best known, not proved: in Friedman's 2009 survey the best lower bound was 6.3506 (Trevor Green), so a smaller square has not been ruled out.
In David W. Cantrell's packing the turned squares stand at about 42.086 and 45 degrees.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 6 + ½√2 ≈ 6.70711 | Frits Göbel | first, adds four L shapes to the 5-square packing |
| early 1980 | 9/2 + (3/2)√2 ≈ 6.62132 | Evert Stenlund | improved |
| 1997 | ≈ 6.62123 | Erich Friedman | improved |
| September 2002 | ≈ 6.59862 | David W. Cantrell | improved, best known today |
| April 2004 | ≈ 6.60323 | Thierry Gensane and Philippe Ryckelynck | did not improve |
Sources
The best known values come from "Squares in Squares", David Ellsworth, from the original by Erich Friedman.
Each size names who holds its value and since when. It is their result that Velmar Cut equals.
Is the packing valid?
Yes. The certificate passes the exact check: 37 pieces, each exactly a unit square, each inside the box, no two overlapping.
How close is it to the best known value?
The side from Velmar Cut minus the best known value is +1.670E-12, within the rule's 1E-11.
A match: the certificate passes the exact check, and |L - best known| <= 1E-11, the difference published unrounded for each size. L is the side of the certificate's box, rounded up at 15 decimals.
Downloads
The sides in the CSV are in units of one square's side. DXF and DWG are in millimetres, with squares of 100 mm.
You don't have to trust us
With plain Python and no other software, verify_exact.py beside the certificate checks the file above by exact arithmetic, with no rounding.
python verify_exact.py 37-squares-in-a-square.csv --squares=37 --reference=6.59861960924436 --epsilon=1e-11It does not prove that the packing is optimal, that the reference is the current best known value, or how the layout was found.
Free trial
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Install Velmar Cut, enter the parts and stock of a current job and print the cutting schedule. The trial starts on first run and needs no account or card.