Hadamard Matrix of Order 668 | Epoch AI

Solution update<br>A solution to this problem has been reported 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.

It is not yet clear whether the full 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”.<br>About the problem<br>A 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.

Mathematicians 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. New cases typically require somewhat clever and novel constructions.

This problem asks for a Hadamard matrix of order \(668\).<br>Prompt

Copy<br>Find a Hadamard matrix of order 668.

Write 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.

Cite

Epoch AI’s work is free to use, distribute, and reproduce provided the source and authors are credited under the Creative Commons Attribution license.<br>Citation<br>Epoch AI (2026), "Hadamard Matrix of Order 668". Published online at epoch.ai. Retrieved from 'https://epoch.ai/frontiermath/open-problems/hadamard' [online resource]. Accessed 12 Aug 2026.

BibTeX Citation<br>@misc{epoch2026,<br>title={Hadamard Matrix of Order 668},<br>author={{Epoch AI}},<br>year={2026},<br>url={https://epoch.ai/frontiermath/open-problems/hadamard},<br>note={Accessed: 2026-08-12}}

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Hadamard Matrix of Order 668<br>"Hadamard Matrix of Order 668" — a problem from FrontierMath: Open Problems, Epoch AI's benchmark of unsolved research mathematics.