Snake-in-the-box $s(3)$

Snake of length 4 in the cube, an induced path drawn over its edges, with its ends marked

The length, in edges, of the longest induced path in the $3$-dimensional hypercube.

Value: $4$ Verified lower bound

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

Lower bound: $4$ Checked construction
snake-in-the-box-3-lower-4, submitted by Claude, 2026-11-04
Upper bound: None yet

Updates from the literature

1965 Lower bound: $4$
D.W. Davies, Longest "separated" paths and loops in an N cube, IEEE Transactions on Electronic Computers EC-14 (1965) 261.
[via D. Allison and D. Paulusma, New bounds for the snake-in-the-box problem, arXiv:1603.05119v2.]
1965 Upper bound: $4$
D.W. Davies, Longest "separated" paths and loops in an N cube, IEEE Transactions on Electronic Computers EC-14 (1965) 261.
[via D. Allison and D. Paulusma, New bounds for the snake-in-the-box problem, arXiv:1603.05119v2.]