Problem dossier · Geometry & Packing
Hadamard's maximal determinant problem
What is the largest possible determinant of an n×n matrix with entries ±1? Hadamard's bound n^(n/2) is attained only at Hadamard orders; other orders are subtler.
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Status
Exact maxima are unknown for infinitely many orders — small unresolved cases persist (n in the 20s–30s and beyond).
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The Angle of Attack
Optimization and exhaustive search over equivalence classes for the smallest unresolved orders; combine with Gram-matrix bounds to certify optima.
Tags: determinant · matrices · optimization · bounds
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The Lab
No instruments built yet. When this problem gets tackled, its interactive instruments — explorers, searches, verifiers running in the browser — live here. See the Collatz dossier for what a fully tackled problem looks like.
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The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
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Related Problems
More open problems in Geometry & Packing and adjacent territory.
- Borsuk's problem in low dimensions — Geometry & Packing
- Kissing numbers in higher dimensions — Geometry & Packing
- Square packing in a square — Geometry & Packing
- Tammes problem — Geometry & Packing
- Hadamard conjecture — Number Theory
- Odd perfect numbers — Number Theory