Explanations

Tutorial

Start with the basic ideas, definitions and examples in the first-principles tutorial. Continue with the advanced tutorial for the methods behind the search and proofs.

Tutorial

Square Packing from First Principles

The problem, bounds, small examples, and how to read a packing result.
Advanced tutorial

Square Packing, Search, and Proof

Linear programs, exact certificates, search, regularization, and global optimality proofs.

Papers

The n=11 series:

  • Part I proves the project’s lower bounds.
  • Part II reviews Overbeck’s certified bound s(11)>31/8 (T-037).
  • Part III reviews Ahmed’s machine-checked proof that Trump’s 1979 packing of eleven squares is optimal (T-060, V3/C3); the review record is pending.

How Record Square Packings Are Found follows the discovery process from a seed through search and local refinement to a certified upper bound, with figures drawn from published record packings.

Part I

New Lower Bounds for Square Packing for n=11

How weighted points and 2-of-3 threshold atoms prove T-018, T-025 and T-026, s(11)≥3.8264…, with interactive figures.
Part II

A Review of the Certified Lower Bound s(11)>31/8 for 11 Squares

Explains Marc Leonard Overbeck’s proof that s(11)>31/8 (T-037): five-site k-of-m charges, k-of-m charges on shrunken parents with strict cores, and a reoptimized certificate over 12,028 angle rows.
Part III

A Review of the Optimality Proof of the Trump Packing of 11 Squares

Explains Manasseh Ahmed’s proof that Trump’s packing is optimal, s(11)=3.8770835… (T-060): construction, case exclusions, capture and local isolation.
Methods tutorial

How Record Square Packings Are Found

How seeds, search, local refinement and exact checks produce record upper bounds.

PDFs

PDFn=1 to 100The first hundred, each labelled with its best-known side and, where the case is open, its strongest verified lower bound.PDFn=1 to 324Every tracked case as its best-known packing, arranged in a triangle on one sheet.

Each poster is also an SVG (1 to 100, 1 to 324), and the atlas README describes them all.

Visuals

Every case from n = 1 to 324, one square at a time: each step draws the best packing known and names the bound it reaches and its source · 8 m 14 s · open the film

A shorter film runs from 1 to 100 (2 m 20 s). Both are on the v0.4.2 release, with the receipt recording each file and the page it was drawn from. Both predate T-060, so at n = 11 they show the lower bound of 28 September, not the proved value. To move the squares yourself, open the Workbench.