Matched: the best known value
Packing 50 unit squares in a square
What is the smallest square that holds 50 squares of side 1, without overlap? The best known answer is a side of 7.5714285714, from Thomas Schadt, optimized by David Ellsworth, 2025. Velmar Cut matched it.
The numbers
| Best known side | 7.5714285714 |
|---|---|
| Best known side, in full | 7.57142857142857 |
| Side from Velmar Cut | 7.5714285714 |
| Side from Velmar Cut, in full | 7.571428571432671 |
| Difference, Velmar Cut minus best known | +4.101E-12 |
| Found by | Thomas Schadt, optimized by David Ellsworth, 2025 |
| Status | Best known, not proved |
| Time to reach it | 656.624 s in all, several runs |
| Threads used | 8 or more |
| Machine | Intel Core i5-12600K, 10 cores, 16 threads |
| Squares | 34 upright, 16 turned |
| A plain grid needs | side 8 |
The time to reach this result on one desktop PC.
The problem at 50 squares
One square more than a grid of side 7 holds, so a plain grid needs side 8. This layout has thirty-four upright squares and sixteen turned, and its side, 7 + 4/7, is a rational number because a 3, 4, 5 triangle, a Pythagorean triple, sets the angle of most of the turned squares. As the source table notes, it is the third packing with a rational side to hold the best known value at its size.
A plain grid: side 8
Best known: side 7.5714285714
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 Thomas Schadt, optimized by David Ellsworth, 2025, and it is their result that Velmar Cut equals.
Best known, not proved: in Friedman's 2009 survey the best lower bound was 7.3174 (Trevor Green), so a smaller square has not been ruled out.
Thomas Schadt found this packing in December 2025 with a program he wrote himself, and David Ellsworth optimized it.
| When | Side | Who | What happened |
|---|---|---|---|
| early 1979 | 7 + ½√2 ≈ 7.70711 | Frits Göbel | first, adds four L shapes to the 10-square packing |
| early 1980 | 11/2 + (3/2)√2 ≈ 7.62132 | Evert Stenlund | improved, adds an L shape to Stenlund's 37-square packing |
| 1997 | ≈ 7.62123 | Erich Friedman | improved, adds an L shape to Friedman's 37-square packing |
| September 2002 | ≈ 7.59862 | David W. Cantrell | improved, together with the 37-square packing it adds an L shape to |
| December 2025 | ≈ 7.57144 | Thomas Schadt | improved |
| date not given | 7 + 4/7 ≈ 7.57143 | David Ellsworth | optimized, best known today |
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: 50 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 +4.101E-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 50-squares-in-a-square.csv --squares=50 --reference=7.57142857142857 --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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