n = 68
Interval verified- Our construction
8.803383074716108386903836683375989697670063329- Cited parent
8.80345993651653- Absolute reduction
0.0000768618004216131
Source contributorsSigvart Brendberg · Thomas Schadt · David Ellsworth
Computational geometry · explicit constructions
Six improved
upper bounds.
New certified constructions for packing congruent unit squares inside a square, at n = 68, 69, 103, 105, 110, and 131.
01
The problem
How small can the containing square be?
s(n) = least side length that can contain n non-overlapping unit squares
A valid construction with side L establishes s(n) ≤ L. It does not, by itself, prove that L is optimal. This release makes construction claims only.
02
The results
Six explicit packings.
Open any figure to inspect or download the complete SVG geometry. Side lengths below are the interval-validated offered sides.
n = 69
Interval verified- Our construction
8.827205507815920656880680735618736931225314831- Cited parent
8.82721205592900- Absolute reduction
0.000006548113079343119
Source contributorsMaurizio Morandi · David W. Cantrell
n = 103
Interval verified- Our construction
10.703790283762427- Cited parent
10.70383477210707- Absolute reduction
0.000044488344643
Source contributorsThomas Schadt · David Ellsworth
n = 105
Interval verified- Our construction
10.807847913867976- Cited parent
10.80789399144854- Absolute reduction
0.000046077580564
Source contributorsThomas Schadt · David Ellsworth
n = 110
Interval verified- Our construction
10.996797773597706- Cited parent
10.99683777797875- Absolute reduction
0.000040004381052
Source contributorsDavid W. Cantrell · David Ellsworth
n = 131
Interval verified- Our construction
11.956543108124773261501115489978287643498579738- Cited parent
11.95654869347733- Absolute reduction
0.0000055853525567384986
Source contributorsKároly Hajba · David Ellsworth