William Echols

Snake-in-the-box updates

Thu, 13 Aug 2026 00:00:00 +0000

This page collects my snake-in-the-box record improvements, along with selected updates from Tom Ace and others for chronological context. It is not intended to be a complete history. For the latest results, I recommend Ace's Snake-in-the-Box lower bounds.

Updates

June 26-29, 2026 · Echols (more details): 12-d snake: 1461→1463; 13-d snake: 2892→2898; 10-d coil: 366→370; 11-d coil: 694→700; 12-d coil: 1346→1352; 13-d coil: 2600→2624; 11-d symmetric coil: 662→674
July 2, 2026 · Echols (more details): 11-d symmetric coil: 674→718; 12-d symmetric coil: 1222→1422; 13-d symmetric coil: 2354→2766
July 8, 2026 · Echols (more details): 13-d snake: 2898→2900; 13-d coil: 2832→2840; 12-d symmetric coil: 1422→1430; 13-d symmetric coil: 2766→2810
July 16, 2026 · Orland et al (A Census of New Snake-in-the-Box Records): 9-d snake: 190→191; 10-d snake: 376→379; 11-d snake: 737→746; 12-d snake: 1465→1476; 13-d snake: 2900→2922; 9-d coil: 188→192; 10-d coil: 370→374; 11-d coil: 728→732; 12-d coil: 1442→1466; 10-d symmetric coil: 362→370; 11-d symmetric coil: 718→726
July 17, 2026 · Ace: 13-d snake: 2922→2928; 11-d coil: 732→734
July 22, 2026 · Echols: 13-d snake: 2928→2932; 11-d coil: 734→736; 12-d coil: 1466→1468; Built from stale 2924- and 732-length starting records, as I was working over ssh and had not loaded Ace's latest results.
July 23, 2026 · Ace: 11-d coil: 736→738
July 23, 2026 · Echols: 11-d coil: 738→740
August 11, 2026 · Ace: 11-d snake: 746→747
August 13, 2026 · Echols: 13-d coil: 2916→2926; 11-d symmetric coil: 726→730; 13-d symmetric coil: 2810→2822; Found by augmenting my approach with a neutral move from Ace.
August 29, 2026 · Echols: 11-d snake: 747→750; 13-d coil: 2926→2930; 12-d symmetric coil: 1430→1434; 13-d symmetric coil: 2822→2830

Last updated September 4, 2026

Improving snakes and coils with further windowed rerouting

Wed, 08 Jul 2026 00:00:00 +0000

Using a modified windowed reouting approach (described here) with the marginal-forbidden heuristic (described here), I was able to find some more improvements to snake-in-the-box, coil-in-the-box, and symmetric coil-in-the-box records.

Findings

Improvements to the 13-d snake-in-the-box construction were marginal, improving my existing length-2898 snake into a length-2900 snake.

New hypercube snake
n	previous-best length	new snake length
13	2898	2900

Ace's 13-d length-2832 coil, first published at www.minortriad.com/snake/, was improved to a length-2840 coil.

Building symmetric coils with implicit-quotient beam search

Thu, 02 Jul 2026 00:00:00 +0000

Before the improvements described here, the following were the best known lengths for the snake-in-the-box, coil-in-the-box, and symmetric coil-in-the-box problems, in dimensions 9 ≤ n ≤ 13: (sourced from www.minortriad.com)

Baseline lengths, as of July 1, 2026
n	snake	coil	symmetric coil
9	190	188	186
10	376	370	362
11	737	728	674
12	1465	1442	1222
13	2898	2832	2354

In dimensions n ≤ 10, the longest known symmetric coils are approximately the same length as the longest known snakes and coils. After the 10th dimension, the lengths of the best-known constructions for these problems deviate significantly, as shown by the percentage of best-known symmetric coil versus best-known (general) coil:

Lengthening hypercube snakes with windowed rerouting

Fri, 26 Jun 2026 00:00:00 +0000

I, like many people, was first introduced to the snake-in-the-box problem through xkcd #3125:

The vertices of an n-th dimensional hypercube can be labelled as n-bit strings. Then, a valid snake is a path such that the (Hamming) distance between consecutive vertices in the path is 1, and the distance between any non-consecutive vertices is at least 2. Our goal is to construct the largest snake possible for a given dimension n.

Prescriptive Analytics for AI-enabled Operations Engineering

Sat, 26 Jul 2025 00:00:00 +0000

This REU led to the publication of "Edge-Enabled Scalable Routing via Graph Neural Network Pruning and Metaheuristic Optimization" at the 2025 ACM/IEEE Symposium on Edge Computing.

Abstract

Edge intelligence enables distributed decision-making by executing complex optimization tasks directly on resource-constrained edge nodes, minimizing reliance on centralized cloud infrastructure. This work introduces a hybrid edge-intelligent routing framework that combines Graph Neural Network (GNN)-based graph pruning with metaheuristic optimization to achieve scalable and low-latency routing at the network edge. The GNN functions as a lightweight inference module that identifies and removes low-utility edges from a sensing network, substantially reducing communication and computational overhead. On the resulting sparse subgraph, a Guided Local Search (GLS) algorithm performs localized route refinement to produce near-optimal paths without central coordination. This integrated GNN-GLS design allows simultaneous graph learning and optimization on edge hardware while retaining solution quality comparable to centralized solvers. Experimental results on synthetic datasets demonstrate a 13.2% runtime improvement on 1000-node graphs with only a 0.7% increase in route length, confirming the feasibility of learning-driven pruning and decentralized metaheuristic search for scalable edge deployment.

175 billion digits of Gamma(1/5)

Sat, 31 May 2025 00:00:00 +0000

The Gamma function is a continuous extension of the factorial function. It is defined for real numbers and complex numbers with the following formula:

$\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} \ \mathrm{d}t, \quad \Re (z) > 0$

It is currently unknown if Gamma(1/5) is irrational or transcendental. If it turned out to be rational, its decimal expansion would eventually fall into a repeating cycle. No such repetition has been found (much less proven), and many believe Gamma(1/5) to most likely be transcendental.

1.5 trillion digits of log(2)

Thu, 09 Sep 2021 00:00:00 +0000

Using y-cruncher, I calculated 1.5 trillion digits of log(2). The hardware used and its information are presented below:

Component	Specification
Processor	2x Intel(R) Xeon(R) CPU E5-2690 v3 @ 2.60GHz
Topology	48 threads / 24 cores / 2 sockets
Usable Memory	252 GiB
Usable Storage	~200 TiB

I used Proxmox to manage an Ubuntu container allocated with the full server’s resources to easily structure and partition the server. Then, to meet Numberworld’s standards for record submission, I ran the calculation twice: