Matched: the best known value
Packing 88 unit squares in a square
What is the smallest square that holds 88 squares of side 1, without overlap? The best known answer is a side of 9.8881530538, from David W. Cantrell and David Ellsworth, 2025. Velmar Cut matched it.
The numbers
| Best known side | 9.8881530538 |
|---|---|
| Best known side, in full | 9.88815305375857 |
| Side from Velmar Cut | 9.8881530538 |
| Side from Velmar Cut, in full | 9.888153053760645 |
| Difference, Velmar Cut minus best known | +2.075E-12 |
| Found by | David W. Cantrell and David Ellsworth, 2025 |
| Status | Best known, not proved |
| Time to reach it | 5491.5599999999995 s in all, several runs |
| Threads used | 6 or more |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 63 upright, 25 turned |
| A plain grid needs | side 10 |
The time to reach this result on one desktop PC.
The problem at 88 squares
Seven squares more than a grid of side 9 holds, so a plain grid needs side 10. The source table gives the best known side exactly, as a root of a polynomial of degree 20 with whole-number coefficients (degree 10 if √2 may appear in them); it is about 0.11 shorter than 10. This layout has sixty-three upright squares and twenty-five turned.
A plain grid: side 10
Best known: side 9.8881530538
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 and David Ellsworth, 2025, and it is their result that Velmar Cut equals.
Best known, not proved optimal: Friedman's 2009 survey gives lower bounds only up to 85 squares. The source page also lists an improvement by Thomas Schadt as pending, so this value may yet be bettered.
David Ellsworth found this packing in November 2024 by adapting and extending David W. Cantrell's improvement for 37 squares of September 2002, and in January 2025 David W. Cantrell and David Ellsworth each improved it on their own. It improves on Erich Friedman's packing of 1997, which extended the 41-square packing Charles F. Cottingham found in 1979.
| When | Side | Who | What happened |
|---|---|---|---|
| 1997 | 17/2 + √2 ≈ 9.914214 | Erich Friedman | found, by extending Charles F. Cottingham's 41-square packing of 1979 |
| August 2002 | ≈ 9.901777 | David W. Cantrell | improved |
| November 2024 | ≈ 9.894448 | David Ellsworth | improved, adapting an improvement David W. Cantrell had found for another size that month |
| November 2024 | ≈ 9.888160 | David Ellsworth | improved, adapting and extending David W. Cantrell's improvement for 37 squares of September 2002 |
| January 2025 | ≈ 9.888153 | David W. Cantrell and David Ellsworth | improved, by each of them independently, best known today |
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: 88 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 +2.075E-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 88-squares-in-a-square.csv --squares=88 --reference=9.88815305375857 --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
Seven days with every edition
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.