Matched: the best known value
Packing 84 unit squares in a square
What is the smallest square that holds 84 squares of side 1, without overlap? The best known answer is a side of 9.7071067812, from Evert Stenlund, 1980. Velmar Cut matched it.
The numbers
| Best known side | 9.7071067812 |
|---|---|
| Best known side, in full | 9.70710678118654 |
| Side from Velmar Cut | 9.7071067812 |
| Side from Velmar Cut, in full | 9.707106781190700 |
| Difference, Velmar Cut minus best known | +4.160E-12 |
| Found by | Evert Stenlund, 1980 |
| Status | Best known, not proved |
| Time to reach it | 13.482 s, one run, not counting a final exactness step |
| Threads used | 8 |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 72 upright, 12 turned |
| A plain grid needs | side 10 |
The time of the run on one desktop PC. A final exactness step came after it and is not counted.
The problem at 84 squares
Three squares more than a grid of side 9 holds, so a plain grid needs side 10. This layout has twelve squares at 45 degrees and seventy-two upright. The side, 9 + ½√2, is exactly one longer than at 67 and has the same form as at 27, 38 and 52.
A plain grid: side 10
Best known: side 9.7071067812
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 Evert Stenlund, 1980, and it is their result that Velmar Cut equals.
Best known, not proved: in Friedman's 2009 survey the best lower bound was 9.2667 (Trevor Green), so a smaller square has not been ruled out.
Evert Stenlund's packing extends the 52-square packing Frits Göbel found in early 1979. Erich Friedman's 2009 survey lists this size under Göbel's name, while the current table credits Stenlund. The same side also comes from adding two L shapes to that 52-square packing.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1980 | 9 + ½√2 ≈ 9.70711 | Evert Stenlund | extends Göbel's 52-square packing, 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: 84 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 +4.160E-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 84-squares-in-a-square.csv --squares=84 --reference=9.70710678118654 --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.
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