Matched: the best known value
Packing 87 unit squares in a square
What is the smallest square that holds 87 squares of side 1, without overlap? The best known answer is a side of 9.8388152699, from Allen Chang, 2026. Velmar Cut matched it.
The numbers
| Best known side | 9.8388152699 |
|---|---|
| Best known side, in full | 9.83881526994826 |
| Side from Velmar Cut | 9.8388152700 |
| Side from Velmar Cut, in full | 9.838815269950145 |
| Difference, Velmar Cut minus best known | +1.885E-12 |
| Found by | Allen Chang, 2026 |
| Status | Best known, not proved |
| Time to reach it | 307.253 s, one run |
| Threads used | 6 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 62 upright, 25 turned |
| A plain grid needs | side 10 |
The time to reach this result on one desktop PC.
The problem at 87 squares
Six squares more than a grid of side 9 holds, so a plain grid needs side 10. The source table gives the best known side as a root of a polynomial of degree 41, about 0.16 shorter than 10; the value is from September 2026. This layout has sixty-two upright squares and twenty-five turned.
A plain grid: side 10
Best known: side 9.8388152699
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 Allen Chang, 2026, and it is their result that Velmar Cut equals.
This side is the best known but not proved optimal; the lower bounds in Friedman's 2009 survey stop at 85 squares, so it gives none for this size.
Allen Chang improved and optimized this packing in September 2026, with help from TheMagicAnimals, and David Ellsworth verified its polynomial solution and completed it with the angles. David Ellsworth had found it in December 2024, basing it on packings of other sizes by Károly Hajba and Joe DeVincentis; David W. Cantrell improved it in January 2025, and Ellsworth improved and optimized it in February 2026.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1980 | 14/3 + (11/3)√2 ≈ 9.852116 | Evert Stenlund | found, similar to the 19-square packing Robert Wainwright found in late 1979 |
| August 2002 | ≈ 9.851975 | David W. Cantrell | improved |
| December 2024 | ≈ 9.846667 | David Ellsworth | improved, based on packings of other sizes by Károly Hajba and Joe DeVincentis |
| January 2025 | ≈ 9.838927 | David W. Cantrell | improved |
| February 2026 | ≈ 9.838859 | David Ellsworth | improved |
| February 2026 | ≈ 9.838817 | David Ellsworth | optimized |
| September 2026 | ≈ 9.838815 | Allen Chang | improved and optimized, with help from TheMagicAnimals, best known today |
| September 2026 | same side | David Ellsworth | verified the polynomial solution and completed it with the angles |
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: 87 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 87-squares-in-a-square.csv --squares=87 --reference=9.83881526994826 --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.
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