Square Packing

Square packing asks for the minimum size square capable of bounding equal squares arranged in any configuration. The first few cases are illustrated above (Friedman). Among the nontrivial cases, packings which have been proven optimal include 2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 22, 23, 24, 33, 34, and 35, in addition to the trivial cases of the square numbers (Friedman and Ellsworth).

If for some , it is conjectured that the size of the minimum bounding square is for small . The smallest for which the conjecture is known to be violated is (with ).

Let be the least side length of a square containing unit squares with arbitrary orientations and pairwise disjoint interiors. Nagamochi (2005) claimed a rectangle bound implying for every integer . Karakus (2026) constructed counterexamples to the scoring assertion used in the proof, so the published proof of the rectangle bound is incomplete, although the bound itself is not disproved.

Karakus (2026) independently proved, for every integer , the weaker result

(1)

and, for every nonsquare integer , the lower bound

(2)

This argument does not establish . Karakus (2026) reports using AI tools during exploratory work, for symbolic and numerical checks, and for language and typesetting assistance, and states that all mathematical arguments and calculations were independently checked.

The following table gives the smallest known side lengths for a square into which unit squares can be packed (Friedman and Ellsworth). An asterisk (*) indicates that a packing has been proven to be optimal.

	exact	approx.		exact	approx.
1*	1	1	16*	4	4
2*	2	2	17		4.6755...
3*	2	2	18		4.822...
4*	2	2	19		4.885...
5*		2.707...	20	5	5
6*	3	3	21	5	5
7*	3	3	22*	5	5
8*	3	3	23*	5	5
9*	3	3	24*	5	5
10*		3.707...	25*	5	5
11		3.877...	26		5.6214...
12	4	4	27		5.7072...
13*	4	4	28		5.8244...
14*	4	4	29		5.9338...
15*	4	4

The value listed for is the polynomial root (Ellsworth).

In December 2025, T. Schadt found a packing of 50 unit squares in a square of side length

(3)

The construction, subsequently optimized by D. Ellsworth, uses orientations determined by a 3, 4, 5 triangle and is the smallest known as of September 2026 for (Ellsworth, Pegg 2026).

The best known packings of squares into a circle are illustrated above for the first few cases (Friedman).

The best known packings of squares into an equilateral triangle are illustrated above for the first few cases (Friedman).

The best packing of a square inside a pentagon, illustrated above, is 1.0673....