Matched: the best known value
Packing 70 unit squares in a square
What is the smallest square that holds 70 squares of side 1, without overlap? The best known answer is a side of 8.8816667570, from Joe DeVincentis, 2014. Velmar Cut matched it.
The numbers
| Best known side | 8.8816667570 |
|---|---|
| Best known side, in full | 8.881666757009 |
| Side from Velmar Cut | 8.8816667570 |
| Side from Velmar Cut, in full | 8.881666757012675 |
| Difference, Velmar Cut minus best known | +3.675E-12 |
| Found by | Joe DeVincentis, 2014 |
| Status | Best known, not proved |
| Time to reach it | 1089.879 s in all, several runs |
| Threads used | 4 or more |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 42 upright, 28 turned |
| A plain grid needs | side 9 |
The time to reach this result on one desktop PC.
The problem at 70 squares
Six squares more than a grid of side 8 holds, so a plain grid needs side 9. The source table gives the best known side as a root of a polynomial of degree 4, about 0.12 shorter than 9. This layout has forty-two upright squares and twenty-eight turned.
A plain grid: side 9
Best known: side 8.8816667570
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 Joe DeVincentis, 2014, and it is their result that Velmar Cut equals.
Best known, not proved: the lower bounds in Friedman's 2009 survey cover 65 to 68 and 79 to 80 but not 70, so it gives none for this size.
Joe DeVincentis found this packing in April 2014. In Erich Friedman's 2009 survey the best known value was still 8.9121, found by David W. Cantrell in 2002.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1980 | 4 + (7/2)√2 ≈ 8.94975 | Evert Stenlund | found |
| 1997 | 15/2 + √2 ≈ 8.91421 | Erich Friedman | improved, extending and adapting the 41-square packing Charles F. Cottingham found in 1979 |
| August 2002 | ≈ 8.91209 | David W. Cantrell | improved |
| April 2014 | ≈ 8.88167 | Joe DeVincentis | improved, best known today |
| November 2024 | ≈ 8.90477 | 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: 70 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 +3.675E-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 70-squares-in-a-square.csv --squares=70 --reference=8.881666757009 --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.