Matched: the best known value
Packing 39 unit squares in a square
What is the smallest square that holds 39 squares of side 1, without overlap? The best known answer is a side of 6.8107220831, from Thomas Schadt, optimized by David Ellsworth, 2026. Velmar Cut matched it.
The numbers
| Best known side | 6.8107220831 |
|---|---|
| Best known side, in full | 6.81072208306864 |
| Side from Velmar Cut | 6.8107220831 |
| Side from Velmar Cut, in full | 6.810722083069860 |
| Difference, Velmar Cut minus best known | +1.220E-12 |
| Found by | Thomas Schadt, optimized by David Ellsworth, 2026 |
| Status | Best known, not proved |
| Time to reach it | 1134.659 s in all, several runs |
| Threads used | 4 or more |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 21 upright, 18 turned |
| A plain grid needs | side 7 |
The time to reach this result on one desktop PC.
The problem at 39 squares
Three squares more than a grid of side 6 holds, so a plain grid needs side 7. As at 28, the source table gives the best known side as a root of a polynomial, here of degree 5. This layout has twenty-one upright squares and eighteen turned. The value is from January 2026.
A plain grid: side 7
Best known: side 6.8107220831
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 Thomas Schadt, optimized by David Ellsworth, 2026, and it is their result that Velmar Cut equals.
This side is the best known but not proved optimal; the best lower bound in Friedman's 2009 survey was 6.3506 (Trevor Green), which leaves room for a smaller square.
Thomas Schadt found this packing in January 2026 with a program he wrote himself, and David Ellsworth refound, refined and optimized it the same month. In Erich Friedman's 2009 survey the best known value was still David W. Cantrell's from 2002.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 4 + 2√2 ≈ 6.82843 | Frits Göbel | first, adds an L shape to the 28-square packing |
| 1997 | 11/2 + ½√7 ≈ 6.82288 | Erich Friedman | improved, extends the alternative 18-square packing Mats Gustafsson found in 1981 |
| August 2002 | ≈ 6.81881 | David W. Cantrell | improved |
| December 2025 | ≈ 6.81829 | Thomas Schadt | improved |
| December 2025 | ≈ 6.81828 | David Ellsworth | optimized |
| January 2026 | ≈ 6.81137 | Thomas Schadt | improved |
| January 2026 | ≈ 6.81072 | David Ellsworth | refound, refined and optimized, 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: 39 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 +1.220E-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 39-squares-in-a-square.csv --squares=39 --reference=6.81072208306864 --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.