Matched: the best known value
Packing 11 unit squares in a square
What is the smallest square that holds 11 squares of side 1, without overlap? The best known answer is a side of 3.8770835900, from Walter Trump, 1979. Velmar Cut matched it.
The numbers
| Best known side | 3.8770835900 |
|---|---|
| Best known side, in full | 3.87708359002281 |
| Side from Velmar Cut | 3.8770835900 |
| Side from Velmar Cut, in full | 3.877083590023390 |
| Difference, Velmar Cut minus best known | +5.800E-13 |
| Found by | Walter Trump, 1979 |
| Status | Best known, not proved |
| Time to reach it | 1.762 s, one run |
| Threads used | 16 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 6 upright, 5 turned |
| A plain grid needs | side 4 |
The time to reach this result on one desktop PC.
The problem at 11 squares
Side 4 is still enough for a plain grid, with room left over. Here the best known side has no concise closed form: it is a root of a polynomial of degree 8. This layout has six upright squares and five turned.
A plain grid: side 4
Best known: side 3.8770835900
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 Walter Trump, 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 3.7888 as the best lower bound (Walter Stromquist), so a smaller square may still exist.
Many people found Walter Trump's packing independently, and the discovery was wrongly attributed to Gustafson and Thule. Its middle squares turn about 40.182 degrees, with a small gap between them. It is rigid: no other valid packing can be reached from it without resizing the enclosing square.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 5/2 + √2 ≈ 3.91421 | Frits Göbel | first |
| 1979 | same side | Charles F. Cottingham | alternative with the fewest turned squares, documented by Martin Gardner in October 1979 |
| 1979 | ≈ 3.87708 | Walter Trump | improved, best known today, rigid |
| 1980 | 2 + (4/3)√2 ≈ 3.88562 | Pertti Hämäläinen | did not improve on the best known value; Walter R. Stromquist proved it in 2002 optimal among packings at 45 degrees |
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: 11 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 +5.800E-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 11-squares-in-a-square.csv --squares=11 --reference=3.87708359002281 --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.
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