what is known, exactly

The best packing and the proven floor

For each n two numbers are known: the side of the best packing anyone has found, and a floor that a proof says no packing can beat. Where they meet, s(n) is settled; elsewhere the truth lies in the gap, and for most n the gap has barely moved in forty years. Floors are collected for n ≤ 100.

The gap

show n up to as a percentage of the side hide n where the floor and the grid agree (settled trivially)

best packing foundproven floorgapsettled (equal)

How one n closed over time

n = August–September 2026 only

best packing (upper bound)proven floor (lower bound)

every recorded floor for n = , with its source

date	floor	by	note

The chart draws each floor on the date it was made public, and only floors that beat every earlier one. The list includes the rest. Others' 2026 certificates are linked as their authors published them; Sources §5 has the details.

Table

all n up to 100

n	best packing	found by	proven floor	proved by	gap	status

Upper bounds and their dates are read from David Ellsworth's catalogue; lower bounds from Erich Friedman's survey and the papers it cites, with the 2026 computer-checked certificates (unrefereed) marked. Bounds of the form ⌈√n⌉ that follow from area alone or from Nagamochi's general theorem are labelled as such.