Matched: the best known value
Packing 40 unit squares in a square
What is the smallest square that holds 40 squares of side 1, without overlap? The best known answer is a side of 6.8284271247, from Frits Göbel, 1979. Velmar Cut matched it.
The numbers
| Best known side | 6.8284271247 |
|---|---|
| Best known side, in full | 6.82842712474619 |
| Side from Velmar Cut | 6.8284271247 |
| Side from Velmar Cut, in full | 6.828427124747490 |
| Difference, Velmar Cut minus best known | +1.300E-12 |
| Found by | Frits Göbel, 1979 |
| Status | Best known, not proved |
| Time to reach it | 0.166 s, one run |
| Threads used | 16 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 24 upright, 16 turned |
| A plain grid needs | side 7 |
The time to reach this result on one desktop PC.
The problem at 40 squares
In this layout sixteen of the forty squares are turned, all at 45 degrees, and the other twenty-four stand upright. That gives 4 + 2√2 against 7 for a plain grid. Like the form at 66, this one has no fractions in it.
A plain grid: side 7
Best known: side 6.8284271247
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 Frits Göbel, 1979, and it is their result that Velmar Cut equals.
The source table lists no proof that this side is optimal: Friedman's 2009 survey gives 6.4061 as the best lower bound (Trevor Green), so a smaller square may still exist.
This is a Göbel square, a construction of Frits Göbel: a 4 by 4 block of squares at 45 degrees in the centre, inside four staircases. Evert Stenlund's first 53-square packing added an L shape to it. It is rigid: unless the enclosing square changes size, it cannot become another valid packing.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 4 + 2√2 ≈ 6.82843 | Frits Göbel | found, best known today, rigid |
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: 40 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 +1.300E-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 40-squares-in-a-square.csv --squares=40 --reference=6.82842712474619 --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.