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Shorter superpermutations on 8, 9 and 10 symbols

New superpermutations shorter than Egan's 2018 bound were posted for 8, 9 and 10 symbols; those for 9 and 10 were developed interactively with GPT-5.6 Sol, which then retrieved them to pass Epoch AI's verifier.

問題
Are there superpermutations on 8, 9 and 10 symbols shorter than Egan's 2018 constructions?
貢献
William Echols with GPT-5.6 Sol (n = 9, 10); Uku Raudvere (n = 8) · Interactive Codex session (n = 9, 10)
検証
Passed Epoch AI's verifier
経緯
Egan 2018 → new constructions 2026-07-26 → verified 2026-07-27
出典
Epoch AI problem page
部分的
結果Improved upper bounds · minimum still open

The problem

A superpermutation on nn symbols is a string that contains every permutation of the nn symbols as a contiguous substring. The shortest possible length is known only for small nn. In 2018 Greg Egan gave short constructions for all nn; Epoch AI's FrontierMath Open Problems asked for shorter ones on 8, 9 and 10 symbols.

What AI did

On 26 July 2026, Uku Raudvere posted a shorter superpermutation for n=8n = 8 (whether he used AI is not stated), and William Echols posted ones for n=9n = 9 and n=10n = 10 that were "developed in an interactive Codex session using GPT-5.6-Sol". On 27 July, GPT-5.6 Sol retrieved these from the posted links and passed Epoch AI's verifier.

How it was checked

A superpermutation can be checked directly by confirming that every permutation occurs in it.

Limitations

These are improved upper bounds; the minimum length for these nn is still unknown. The AI's part is clearest for n=9n = 9 and n=10n = 10; for the benchmark itself the model only retrieved existing posts. Problem description paraphrased from Epoch AI (CC BY 4.0).

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