Matched: the best known value
Packing 41 unit squares in a square
What is the smallest square that holds 41 squares of side 1, without overlap? The best known answer is a side of 6.9266930945, from Thomas Schadt, optimized by David Ellsworth, 2026. Velmar Cut matched it.
The numbers
| Best known side | 6.9266930945 |
|---|---|
| Best known side, in full | 6.9266930944688 |
| Side from Velmar Cut | 6.9266930945 |
| Side from Velmar Cut, in full | 6.926693094470345 |
| Difference, Velmar Cut minus best known | +1.545E-12 |
| Found by | Thomas Schadt, optimized by David Ellsworth, 2026 |
| Status | Best known, not proved |
| Time to reach it | 1660.447 s in all, several runs |
| Threads used | 4 or more |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 26 upright, 15 turned |
| A plain grid needs | side 7 |
The time to reach this result on one desktop PC.
The problem at 41 squares
Five squares more than a grid of side 6 holds, so a plain grid needs side 7. The source table gives the best known side as a root of a polynomial of degree 42, far longer than at 28 or 39. This layout has twenty-six upright squares and fifteen turned.
A plain grid: side 7
Best known: side 6.9266930945
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.
The source table lists no proof that this side is optimal: Friedman's 2009 survey gives 6.4061 as the best lower bound (Trevor Green), so a smaller square may still exist.
Thomas Schadt found this packing in December 2025 with a program he wrote himself. David Ellsworth refound and refined it around the turn of the year and optimized it in January 2026. Erich Friedman's 2009 survey notes that Charles F. Cottingham's packing of 1979 was only improved in 2005, by David W. Cantrell.
| When | Side | Who | What happened |
|---|---|---|---|
| 1979 | 2 + (7/2)√2 ≈ 6.94975 | Charles F. Cottingham | found |
| early 1980 | 11/3 + (7/3)√2 ≈ 6.96650 | Evert Stenlund | did not improve |
| October 2005 | ≈ 6.94725 | David W. Cantrell | improved |
| April 2014 | ≈ 6.93787 | Joe DeVincentis | improved |
| November 2024 | ≈ 6.94614 | David Ellsworth | did not improve on the best known value |
| December 2025 | ≈ 6.93669 | Thomas Schadt | improved |
| December 2025 | ≈ 6.93667 | David Ellsworth | optimized |
| December 2025 | ≈ 6.92694 | Thomas Schadt | improved |
| December 2025 to January 2026 | ≈ 6.92670 | David Ellsworth | refound and refined |
| January 2026 | ≈ 6.92669 | David Ellsworth | 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: 41 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 +1.545E-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 41-squares-in-a-square.csv --squares=41 --reference=6.9266930944688 --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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