Matched: the best known value
Packing 27 unit squares in a square
What is the smallest square that holds 27 squares of side 1, without overlap? The best known answer is a side of 5.7071067812, from Frits Göbel, 1979. Velmar Cut matched it.
The numbers
| Best known side | 5.7071067812 |
|---|---|
| Best known side, in full | 5.70710678118654 |
| Side from Velmar Cut | 5.7071067812 |
| Side from Velmar Cut, in full | 5.707106781187605 |
| Difference, Velmar Cut minus best known | +1.065E-12 |
| Found by | Frits Göbel, 1979 |
| Status | Best known, not proved |
| Time to reach it | 0.837 s, one run |
| Threads used | 16 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 25 upright, 2 turned |
| A plain grid needs | side 6 |
The time to reach this result on one desktop PC.
The problem at 27 squares
Only two squares need to turn here. This layout keeps twenty-five upright and sets the other two at 45 degrees, and the side comes down from the plain grid's 6 to 5 + ½√2. It is the form seen at 5 and 10, and seen again at 38, 52, 67 and 84.
A plain grid: side 6
Best known: side 5.7071067812
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.
This side is the best known but not proved optimal; the best lower bound in Friedman's 2009 survey was 5.3918 (Trevor Green), which leaves room for a smaller square.
This is a strip packing by Frits Göbel: a strip of squares one wide at 45 degrees in the centre, between two staircases of upright squares. These strips stay the best known at every step David Ellsworth lists up to a very large size, except the one at 17 squares.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 5 + ½√2 ≈ 5.70711 | Frits Göbel | found, best known today |
| 2005 | same side | David W. Cantrell | alternative with the fewest turned squares |
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: 27 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 +1.065E-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 27-squares-in-a-square.csv --squares=27 --reference=5.70710678118654 --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.