n = 324 · Case Records · The Squares Project

n = 324 provedO=R

$s (324) = 18$

The best packing known for 324 squares, side

18

171819

\sqrt{n}

\sqrt{n} + 1

Proven

$s (324) = 18$

optimal
exact
rigid

Bounds

Best known packing

$18$

Construction: grid
Tilt angles: $0^{\circ}$
Source: [Kingbird]
Evidence: E-kingbird-grid-completeness

Verified upper bound

$18$

The reported value, verified here.

Evidence: E-basic-grid-upper

Reported lower bound

Kind: perfect square
Evidence: E-basic-area-lower

Verified lower bound

$18$

The reported value, verified here.

Evidence: E-basic-area-lower

Gap

Solved: the verified bounds meet.

Results in the register

T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
T-085 V3 C3 Karakuş; chelokot · 2026-10-02 · 315 cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$

Verification

upper: replayed here; lower: replayed here

—

Rigidity

locally rigid, verified, exact algebraic

Evidence: E-perfect-square-tiling-rigid

Scope

This record's side is verified above and below at exactly 18, so its 324 unit squares of total area 324 sit in a 18 by 18 container of area 324. The packing is a tiling with no slack anywhere, so no square admits any feasible motion. This is global rather than local, and holds for the exact configuration rather than for a materialized approximation of it. It says nothing about s(n) beyond what the tiling itself shows.

Evidence and sources

4 evidence entries

E-kingbird-grid-completeness, E-basic-area-lower, E-basic-grid-upper, E-nagamochi-lower

[Kingbird] record catalogue
[Friedman DS7] survey

In the frontier survey On GitHub

$s (324)$ — solved

$s (324) = 18$ . Established by triviality (a perfect square).

The packing

The record catalogue does not picture $n = 324$ : no arrangement has ever been found that beats the trivial $⌈ \sqrt{324} ⌉ = 18$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

$324$ is a perfect square, so the $18 \times 18$ grid is optimal and the area bound $\sqrt{n}$ is already tight.

Corrected 4 October 2026: the verified lower bound cites the area bound, E-basic-area-lower; until 2 October it cited Nagamochi 2005 (E-nagamochi-lower) at the same value, whose Lemma 1 is false (D-516).

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.

bound	evidence	run	code	programs
verified lower	`E-basic-area-lower`	replayed here	independent	`V-check-basic-bounds` (first-party)
verified upper	`E-basic-grid-upper`	replayed here	independent	`V-check-basic-bounds` (first-party)

Bounds

Results in the register

\(s (324)\)s(324) — solved

The packing

The lower bound

Verification Code

$s (324)$ — solved