Matched: the best known value
Packing 18 unit squares in a square
What is the smallest square that holds 18 squares of side 1, without overlap? The best known answer is a side of 4.8228756555, from Pertti Hämäläinen, 1980. Velmar Cut matched it.
The numbers
| Best known side | 4.8228756555 |
|---|---|
| Best known side, in full | 4.82287565553229 |
| Side from Velmar Cut | 4.8228756555 |
| Side from Velmar Cut, in full | 4.822875655533110 |
| Difference, Velmar Cut minus best known | +8.200E-13 |
| Found by | Pertti Hämäläinen, 1980 |
| Status | Best known, not proved |
| Time to reach it | 0.471 s, one run |
| Threads used | 16 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 12 upright, 6 turned |
| A plain grid needs | side 5 |
The time to reach this result on one desktop PC.
The problem at 18 squares
One square more than at 17, and the side has a closed form again: 7/2 + ½√7, with √7 in place of the more usual √2. A plain grid still needs side 5. This layout has twelve upright squares and six turned, and the same form returns at 53.
A plain grid: side 5
Best known: side 4.8228756555
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 Pertti Hämäläinen, 1980, 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 4.4452 (Trevor Green), which leaves room for a smaller square.
In the packings of Pertti Hämäläinen and Mats Gustafsson the middle squares turn about 24.295 degrees.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 2 + 2√2 ≈ 4.82843 | Frits Göbel | first |
| 1980 | 7/2 + ½√7 ≈ 4.82288 | Pertti Hämäläinen | improved, best known today |
| 1981 | same side | Mats Gustafsson | alternative with the fewest turned squares |
| September 2002 | same side | David W. Cantrell | alternative |
| April 2004 | same side | Thierry Gensane and Philippe Ryckelynck | alternative |
| December 2024 | 2 + 2√2 ≈ 4.82843 | David Ellsworth | rigid alternative at Göbel's side, did not improve |
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: 18 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 +8.200E-13, 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 18-squares-in-a-square.csv --squares=18 --reference=4.82287565553229 --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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