Matched: the best known value
Packing 89 unit squares in a square
What is the smallest square that holds 89 squares of side 1, without overlap? The best known answer is a side of 9.9497474683, from Evert Stenlund, 1980. Velmar Cut matched it.
The numbers
| Best known side | 9.9497474683 |
|---|---|
| Best known side, in full | 9.94974746830583 |
| Side from Velmar Cut | 9.9497474683 |
| Side from Velmar Cut, in full | 9.949747468308645 |
| Difference, Velmar Cut minus best known | +2.815E-12 |
| Found by | Evert Stenlund, 1980 |
| Status | Best known, not proved |
| Time to reach it | 96.19 s, one run |
| Threads used | 8 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 40 upright, 49 turned |
| A plain grid needs | side 10 |
The time to reach this result on one desktop PC.
The problem at 89 squares
Eight squares more than a grid of side 9 holds, so a plain grid needs side 10. The best known side, 5 + (7/2)√2, is only about 0.05 shorter. This layout has more squares turned than upright: forty-nine turned and forty upright.
A plain grid: side 10
Best known: side 9.9497474683
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 Evert Stenlund, 1980, and it is their result that Velmar Cut equals.
Best known, not proved: the lower bounds in Friedman's 2009 survey stop at 85 squares, so it gives none for this size.
Evert Stenlund found this packing in early 1980, continuing a pattern Frits Göbel found in early 1979. Friedman's 2009 survey already lists this value under Stenlund's name.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1980 | 5 + (7/2)√2 ≈ 9.94975 | Evert Stenlund | found, continuing a pattern of Frits Göbel's from early 1979, best known today |
| June 2023 | same side | David Ellsworth | rearrangement with the fewest turned squares |
| January 2026 | ≈ 9.96543 | David Ellsworth | did not improve on the best known value |
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: 89 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 +2.815E-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 89-squares-in-a-square.csv --squares=89 --reference=9.94974746830583 --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.