Hadamard Matrices

Find a Hadamard matrix of order 668

Notability: Moderately interesting

Solved: No

Field: Combinatorics

Contributor: Epoch Staff

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About the problem Attempts by AI AI prompts Mathematician survey

About the problem

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 $668$. As a warm-up, we ask for a matrix of order $428$.

Attempts by AI

We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.

Model

Problem

Solved

Logs

GPT-5.2 Pro

Warm-up

Yes

GPT-5.2 Pro

Full Problem

Gemini 3 Deep Think

Full Problem

AI Prompts

Warm-up

Find a Hadamard matrix of order 428. Provide your solution as a csv.

Full problem

Find a Hadamard matrix of order 668. Provide your solution as a csv.

Mathematician survey

The author assessed the problem as follows.

# of mathematicians highly familiar with the problem: a majority of those working in a subfield (≈100)

# of mathematicians who have made a serious attempt to solve the problem: 5–10

Rough guess of how long it would take an expert human to solve the problem: 1–4 weeks

Notability of a solution: moderately interesting

A solution would be published: in a standard specialty journal

Likelihood of a solution generating more interesting math: somewhat likely: an especially innovative solution could, but is hardly guaranteed

Probability that the problem is solvable as stated: 95–99%