Matched: the best known value
Packing 53 unit squares in a square
What is the smallest square that holds 53 squares of side 1, without overlap? The best known answer is a side of 7.8228756555, from David Ellsworth, 2026. Velmar Cut matched it.
The numbers
| Best known side | 7.8228756555 |
|---|---|
| Best known side, in full | 7.82287565553229 |
| Side from Velmar Cut | 7.8228756555 |
| Side from Velmar Cut, in full | 7.822875655533935 |
| Difference, Velmar Cut minus best known | +1.645E-12 |
| Found by | David Ellsworth, 2026 |
| Status | Best known, not proved |
| Time to reach it | 86.758 s, one run |
| Threads used | 8 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 33 upright, 20 turned |
| A plain grid needs | side 8 |
The time to reach this result on one desktop PC.
The problem at 53 squares
One square more than at 52 calls for a very different layout: this one has twenty turned squares against eight at 52, and thirty-three upright. A plain grid still needs side 8. The side, 13/2 + ½√7, is the form from 18 again, with 13/2 in place of 7/2.
A plain grid: side 8
Best known: side 7.8228756555
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 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 7.3174 (Trevor Green), which leaves room for a smaller square.
David Ellsworth holds today's value, from 2026. It came after four improvements in a row on this one size, three of them since 2024.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1980 | 5 + 2√2 ≈ 7.82843 | Evert Stenlund | adds an L shape to Göbel's 40-square packing |
| September 2002 | ≈ 7.82304 | David W. Cantrell | improved |
| December 2024 | ≈ 7.82291 | David W. Cantrell | improved |
| February 2026 | ≈ 7.822883 | David Ellsworth | improved |
| February 2026 | 13/2 + ½√7 ≈ 7.822876 | 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: 53 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 +1.645E-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 53-squares-in-a-square.csv --squares=53 --reference=7.82287565553229 --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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