Matched: the best known value
Packing 69 unit squares in a square
What is the smallest square that holds 69 squares of side 1, without overlap? The best known answer is a side of 8.8271946557, from hmbelvedere, optimized by David Ellsworth, 2026. Velmar Cut matched it.
The numbers
| Best known side | 8.8271946557 |
|---|---|
| Best known side, in full | 8.82719465572973 |
| Side from Velmar Cut | 8.8271946557 |
| Side from Velmar Cut, in full | 8.827194655731615 |
| Difference, Velmar Cut minus best known | +1.885E-12 |
| Found by | hmbelvedere, optimized by David Ellsworth, 2026 |
| Status | Best known, not proved |
| Time to reach it | 135.027 s, one run |
| Threads used | 6 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 50 upright, 19 turned |
| A plain grid needs | side 9 |
The time to reach this result on one desktop PC.
The problem at 69 squares
Five squares past a grid of side 8, so a plain grid needs side 9. The source table gives the best known side as a root of a polynomial of degree 38, about 0.17 shorter than 9, and the value dates from September 2026. This layout has fifty upright squares and nineteen turned.
A plain grid: side 9
Best known: side 8.8271946557
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 hmbelvedere, optimized by David Ellsworth, 2026, and it is their result that Velmar Cut equals.
Best known, not proved: the lower bounds in Friedman's 2009 survey cover 65 to 68 and 79 to 80 but not 69, so it gives none for this size.
In July 2026 hmbelvedere improved this packing, and in September 2026 David Ellsworth optimized it. It goes back to Maurizio Morandi's packing of June 2010, which David W. Cantrell improved in August 2023.
| When | Side | Who | What happened |
|---|---|---|---|
| 1997 | 5/2 + (9/2)√2 ≈ 8.863961 | Erich Friedman | found |
| September 2002 | ≈ 8.856180 | David W. Cantrell | improved |
| June 2010 or a little earlier | ≈ 8.851093 | Maurizio Morandi | improved, apparently by him |
| June 2010 or a little earlier | ≈ 8.848790 | Maurizio Morandi | improved, adding an L shape to a packing by David W. Cantrell |
| June 2010 | ≈ 8.828687 | Maurizio Morandi | improved |
| August 2023 | ≈ 8.827212 | David W. Cantrell | improved |
| November 2024 | ≈ 8.841165 | David Ellsworth | did not improve on the best known value |
| July 2026 | ≈ 8.827205 | hmbelvedere | improved |
| September 2026 | ≈ 8.827208 | David Ellsworth | refound |
| September 2026 | ≈ 8.827195 | 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: 69 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 +1.885E-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 69-squares-in-a-square.csv --squares=69 --reference=8.82719465572973 --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.