Matched: the best known value
Packing 86 unit squares in a square
What is the smallest square that holds 86 squares of side 1, without overlap? The best known answer is a side of 9.8228756555, from Erich Friedman, 1997. Velmar Cut matched it.
The numbers
| Best known side | 9.8228756555 |
|---|---|
| Best known side, in full | 9.82287565553229 |
| Side from Velmar Cut | 9.8228756555 |
| Side from Velmar Cut, in full | 9.822875655534500 |
| Difference, Velmar Cut minus best known | +2.210E-12 |
| Found by | Erich Friedman, 1997 |
| Status | Best known, not proved |
| Time to reach it | 406.197 s in all, several runs |
| Threads used | 6 or more |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 63 upright, 23 turned |
| A plain grid needs | side 10 |
The time to reach this result on one desktop PC.
The problem at 86 squares
Five squares more than a grid of side 9 holds, so a plain grid needs side 10. The side has a closed form, 17/2 + ½√7: the form from 18 and 53 once more, now with 17/2 in front. This layout has sixty-three upright squares and twenty-three turned.
A plain grid: side 10
Best known: side 9.8228756555
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 Erich Friedman, 1997, and it is their result that Velmar Cut equals.
The source table lists no proof that this side is optimal, and Friedman's 2009 survey has no lower bound for it: its lower bounds stop at 85 squares.
Erich Friedman's packing of 1997 extends the alternative that Mats Gustafsson found in 1981 to Pertti Hämäläinen's 18-square packing of 1980. It improved on Evert Stenlund's packing of early 1980, and its value has stood since.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1980 | 7 + 2√2 ≈ 9.82843 | Evert Stenlund | found |
| 1997 | 17/2 + ½√7 ≈ 9.82288 | Erich Friedman | improved, 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: 86 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 +2.210E-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 86-squares-in-a-square.csv --squares=86 --reference=9.82287565553229 --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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