Matched: the best known value
Packing 66 unit squares in a square
What is the smallest square that holds 66 squares of side 1, without overlap? The best known answer is a side of 8.6568542495, from Evert Stenlund, 1980. Velmar Cut matched it.
The numbers
| Best known side | 8.6568542495 |
|---|---|
| Best known side, in full | 8.65685424949238 |
| Side from Velmar Cut | 8.6568542495 |
| Side from Velmar Cut, in full | 8.656854249494200 |
| Difference, Velmar Cut minus best known | +1.820E-12 |
| Found by | Evert Stenlund, 1980 |
| Status | Best known, not proved |
| Time to reach it | 15.243 s, one run |
| Threads used | 8 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 40 upright, 26 turned |
| A plain grid needs | side 9 |
The time to reach this result on one desktop PC.
The problem at 66 squares
Add one square to 65 and the form of the side changes completely, to 3 + 4√2, while a plain grid stays at side 9. This layout has forty upright squares, as at 65, and twenty-six turned.
A plain grid: side 9
Best known: side 8.6568542495
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: in Friedman's 2009 survey the best lower bound was 8.2899 (Trevor Green), so a smaller square has not been ruled out.
Evert Stenlund's packing uses a diagonal strip of width 3, whose squares touch only the squares at two opposite corners. Stenlund improved many of Charles F. Cottingham's packings.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1980 | 3 + 4√2 ≈ 8.65685 | Evert Stenlund | found, best known today |
| 2023 | same side | David W. Cantrell | alternative with rotational symmetry |
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: 66 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 +1.820E-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 66-squares-in-a-square.csv --squares=66 --reference=8.65685424949238 --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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