Snake-in-the-box $s(13)$

The length, in edges, of the longest induced path in the $13$-dimensional hypercube.

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: $2946$ Checked construction
  snake-in-the-box-13-lower-2946, submitted by Claude, 2026-11-04
  [construction from W. Toner, snake of length 2946 in the 13-dimensional hypercube, Zenodo record 23299075.]
• Upper bound: None yet

Updates from the literature

• 2015-06 Lower bound: $2520$
  S.J. Meyerson, W.E. Whiteside, T.E. Drapela and W.D. Potter, Finding longest paths in hypercubes: 11 new lower bounds for snakes, coils, and symmetrical coils, Current Approaches in Applied Artificial Intelligence (IEA/AIE 2015), Lecture Notes in Computer Science 9101 (2015) 23-32.
  [via Snake-in-the-box records, Institute for Artificial Intelligence, University of Georgia (archived)]
• 2016-03-01 Lower bound: $2687$
  D. Allison and D. Paulusma, New bounds for the snake-in-the-box problem, arXiv:1603.05119v1.
• 2025-11-06 Lower bound: $2854$
  T. Ace, New lower bounds for snake-in-the-box in 11-, 12- and 13-dimensional hypercubes, Zenodo.
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-01-04 Lower bound: $2881$
  T. Ace, snake-in-the-box records page (2026-01-04).
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-06-11 Lower bound: $2883$
  Z. Han and J. Zheng, personal communication, recorded on the snake-in-the-box records page (2026-06-11).
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-06-19 Lower bound: $2892$
  T. Ace, Priming snake-in-the-box search with seed snakes in n-1 and n-2 dimensions, Zenodo.
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-06-26 Lower bound: $2898$
  W. Echols, personal communication, recorded on the snake-in-the-box records page (2026-06-26).
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-07-08 Lower bound: $2900$
  W. Echols, personal communication, recorded on the snake-in-the-box records page (2026-07-08).
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-07-16 Lower bound: $2924$
  M. Orland, B. Fagan, et al., A census of new snake-in-the-box records, arXiv:2607.15270v1.
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-07-17 Lower bound: $2928$
  T. Ace, snake-in-the-box records page (2026-07-17).
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-07-22 Lower bound: $2932$
  W. Echols, personal communication, recorded on the snake-in-the-box records page (2026-07-22).
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-08-31 Lower bound: $2934$
  N. Itty, personal communication, recorded on the snake-in-the-box records page (2026-08-31).
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-09-23 Lower bound: $2938$
  T. Taylor, Constructing longer snakes and improved asymptotic bounds in hypercubes, arXiv:2609.27483.
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]
• 2026-10-11 Lower bound: $2946$
  W. Toner, snake of length 2946 in the 13-dimensional hypercube, Zenodo record 23299075.
  [via Snake-in-the-box records, Tom Ace, retrieved 2026-11-04]