{"slug": "hadamard-matrix-of-order-668", "title": "Hadamard Matrix of Order 668", "summary": "A researcher from Anthropic reported a solution to the Hadamard matrix problem for order 668, crediting a team of three humans and Claude. The solution encodes Hadamard matrices for all previously unknown admissible orders up to 2000, including 668, and has been provisionally marked as solved by AI. It remains unclear whether the full Hadamard conjecture has been resolved.", "body_md": "## Solution update\n\nA solution to this problem has been [reported](https://x.com/__alpoge__/status/2087504785952182273?s=20) by a researcher from Anthropic, crediting a team of three humans and Claude. The solution — posted in the form of a puzzle — encodes Hadamard matrices of all admissible orders up to 2000 where no such instance was previously known. This includes one of order 668. We have provisionally marked the problem as being solved by AI, though will revise this if further reports from the team indicate that humans contributed significantly to the core mathematical ideas.\n\nIt is not yet clear whether the full [Hadamard conjecture](https://en.wikipedia.org/wiki/Hadamard_matrix#Hadamard_conjecture) — asking whether a Hadamard matrix exists for *all* admissible orders — has been resolved. It is possible that this result comes from an improved search strategy, or a fully general construction. A resolution to the full conjecture would, a priori, likely have rated as higher than “Moderately Interesting”.\n\n## About the problem\n\nA [Hadamard matrix](https://en.wikipedia.org/wiki/Hadamard_matrix) is a square matrix all of whose entries are \\(\\pm 1\\) and whose rows are mutually orthogonal. Beyond trivial cases, the order of such a matrix must be a multiple of four. The Hadamard conjecture states that a Hadamard matrix exists for every such order. It remains open.\n\nMathematicians have chipped away at finding examples of Hadamard matrices of larger and larger order. The smallest case for which no matrix is known is \\(668\\). The previous smallest unknown case was \\(428\\), resolved in 2004 by [Kharaghani and Tayfeh-Rezaie](https://onlinelibrary.wiley.com/doi/10.1002/jcd.20043). New cases typically require somewhat clever and novel constructions.\n\nThis problem asks for a Hadamard matrix of order \\(668\\).\n\n## Prompt\n\n```\nFind a Hadamard matrix of order 668.\n\nWrite your Hadamard matrix to an .npy file (using `numpy.save`) and output the filepath. The matrix must be a square array of +1 and -1 entries.\n```\n\n", "url": "https://wpnews.pro/news/hadamard-matrix-of-order-668", "canonical_source": "https://epoch.ai/frontiermath/open-problems/hadamard", "published_at": "2026-08-12 17:31:27+00:00", "updated_at": "2026-08-12 17:41:47.227130+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research"], "entities": ["Anthropic", "Claude", "Kharaghani", "Tayfeh-Rezaie"], "alternates": {"html": "https://wpnews.pro/news/hadamard-matrix-of-order-668", "markdown": "https://wpnews.pro/news/hadamard-matrix-of-order-668.md", "text": "https://wpnews.pro/news/hadamard-matrix-of-order-668.txt", "jsonld": "https://wpnews.pro/news/hadamard-matrix-of-order-668.jsonld"}}