Matched: the best known value
Packing 19 unit squares in a square
What is the smallest square that holds 19 squares of side 1, without overlap? The best known answer is a side of 4.8856180832, from Robert Wainwright, 1979. Velmar Cut matched it.
The numbers
| Best known side | 4.8856180832 |
|---|---|
| Best known side, in full | 4.88561808316412 |
| Side from Velmar Cut | 4.8856180832 |
| Side from Velmar Cut, in full | 4.885618083164925 |
| Difference, Velmar Cut minus best known | +8.050E-13 |
| Found by | Robert Wainwright, 1979 |
| Status | Best known, not proved |
| Time to reach it | 13.528 s, one run |
| Threads used | 16 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 10 upright, 9 turned |
| A plain grid needs | side 5 |
The time to reach this result on one desktop PC.
The problem at 19 squares
As at 17 and 18, a plain grid needs side 5. The best known side is 3 + (4/3)√2. This layout has ten upright squares and nine turned, close to an even split.
A plain grid: side 5
Best known: side 4.8856180832
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 Robert Wainwright, 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 4.4852 as the best lower bound (Erich Friedman), so a smaller square may still exist.
Robert Wainwright's packing is based on the diagonal strips Charles F. Cottingham had introduced earlier in 1979, and the value has stood since.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 4 + (2/3)√2 ≈ 4.94281 | Frits Göbel | first |
| early 1979 | 7/2 + √2 ≈ 4.91421 | Charles F. Cottingham | improved, the first to use diagonal strips of width 2 |
| late 1979 | 3 + (4/3)√2 ≈ 4.88562 | Robert Wainwright | improved, best known today; the first of many who found it independently that year |
| 1984 | ≈ 4.88811 | Walter R. Stromquist | did not improve |
| 2002 | same side as today | David W. Cantrell | alternatives, some 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: 19 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 +8.050E-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 19-squares-in-a-square.csv --squares=19 --reference=4.88561808316412 --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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