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arxiv: 2605.04603 · v2 · submitted 2026-05-06 · 🧮 math.CO

Nonexistence of Whirling-Knight Tours at Half Coil Count for n equiv 4, 6 pmod 8

Shisheng Li This is my paper

Pith reviewed 2026-05-12 04:11 UTC · model grok-4.3

classification 🧮 math.CO
keywords whirling knight tourscoil countnonexistenceFarkas certificatesHamiltonian cyclescycle-cover LPmodular arithmeticknight graphs
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The pith

No whirling knight's tour with coil count exactly n/2 exists on boards where n ≡ 4 or 6 mod 8.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that whirling knight's tours with coil count c equal to half the board size are impossible when the board dimension n satisfies n congruent to 4 modulo 8 for n at least 4, or congruent to 6 modulo 8 for n at least 6. A whirling knight's tour is a closed path that visits every square of an n by n board exactly once using only counter-clockwise knight moves, with the coil count recording the number of full windings around the center. The authors supply explicit algebraic certificates that demonstrate the impossibility for each of the two families of board sizes. The result confirms a prior conjecture and shows that the obstruction is uniform across all qualifying board sizes rather than sporadic. Mapping out these forbidden winding numbers narrows the set of achievable configurations for knight paths on square grids.

Core claim

No such tour with c = n/2 exists when n ≡ 4 (mod 8) (n ≥ 4) or n ≡ 6 (mod 8) (n ≥ 6). For each residue class the authors exhibit a closed-form Farkas certificate for infeasibility of a cycle-cover LP relaxation; the two certificates are structurally distinct.

What carries the argument

Closed-form Farkas certificates for the infeasibility of the cycle-cover linear programming relaxation of the whirling knight tour problem in the directed knight graph.

If this is right

  • No half-coil whirling knight's tours exist on any board of the stated sizes, not only on small examples.
  • Structurally different certificates are required for the two congruence classes because their infeasibility proofs differ.
  • The conjecture that half-coil whirling tours are impossible in these cases is settled for all qualifying n.
  • The linear programming certificates provide a uniform algebraic proof that applies to infinitely many board sizes at once.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same certificate technique could be tested on other fixed coil counts besides n/2 to map the full set of attainable windings.
  • The modular conditions on n suggest that achievable coil counts are constrained by arithmetic properties of the board size.
  • One could ask whether the existence question for the complementary residue classes (n ≡ 0, 2 mod 8) admits similar certificate-based resolutions.

Load-bearing premise

The cycle-cover LP relaxation must be a faithful model of the actual tours, so that infeasibility of the relaxation implies there is no real Hamiltonian cycle satisfying the winding condition.

What would settle it

An explicit construction of one whirling knight's tour with coil count exactly n/2 on any single board whose size is congruent to 4 or 6 modulo 8 would disprove the nonexistence claim.

Figures

Figures reproduced from arXiv: 2605.04603 by Shisheng Li.

Figure 1
Figure 1. Figure 1: Coordinates on a 4 × 4 board, pivot at (1.5, 1.5). 2 view at source ↗
Figure 2
Figure 2. Figure 2: The dashed segment is the north plumb-line above the pivot. The arrow is the CCW view at source ↗
Figure 4
Figure 4. Figure 4: The Farkas certificate (α = 1H, β = 0, γ = −1) for (LP3,2). The three red arcs of T3 each cross the north plumb￾line; coil count 3 ̸= 2 = c, witnessing infeasi￾bility. For (a) take w = v = (i, h − 1), so jv − pj = − 1 2 and iv − pi = i − (h − 1 2 ) ≤ −3 2 (since i ≤ h − 2). Plugging each of the eight knight steps into (3) shows that exactly the four steps ∆ ∈ {(1, −2),(−1, −2),(2, −1),(−2, −1)} are CCW (th… view at source ↗
Figure 5
Figure 5. Figure 5: The certificate support of Theorem 1 for view at source ↗
Figure 6
Figure 6. Figure 6: α-support of certificate (8) at n = 12, m = 1. Light-blue cells (Teven) carry αv = −1; dark-blue cells in column h − 1 = 5 at rows R = {0, 4} carry αv = +1. beta-support view at source ↗
read the original abstract

A whirling knight's tour is a Hamiltonian cycle in the digraph of counter-clockwise knight steps about the centre of an $n \times n$ board; its coil count $c$ is the winding number around the centre. We prove that no such tour with $c = n/2$ exists when $n \equiv 4 \pmod 8$ ($n \ge 4$) or $n \equiv 6 \pmod 8$ ($n \ge 6$), settling a conjecture of Beluhov. For each residue class we exhibit a closed-form Farkas certificate for infeasibility of a cycle-cover LP relaxation; the two certificates are structurally distinct.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper proves that no whirling-knight tour with coil count c = n/2 exists on an n×n board when n ≡ 4 (mod 8) for n ≥ 4 or n ≡ 6 (mod 8) for n ≥ 6. This is achieved by constructing closed-form Farkas certificates that certify the infeasibility of a cycle-cover LP relaxation consisting of degree constraints and a winding-number constraint for c = n/2.

Significance. The result settles a conjecture of Beluhov. The provision of explicit algebraic certificates that can be verified by direct substitution into the dual LP is a notable strength, as it allows independent checking without relying on computational search or fitted parameters. The structural distinction between the certificates for the two congruence classes further bolsters the proof's reliability.

minor comments (1)
  1. The ranges n ≥ 4 and n ≥ 6 are stated clearly in the abstract but could be reiterated explicitly when the LP is first defined to aid readers who begin with the technical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, for accurately summarizing the main result, and for recommending acceptance. The referee's assessment of the significance of the explicit Farkas certificates is particularly appreciated.

Circularity Check

0 steps flagged

No significant circularity; direct Farkas certificate on explicit LP

full rationale

The derivation defines an explicit cycle-cover LP (degree constraints plus one linear winding-number equation for c = n/2) and exhibits closed-form Farkas certificates that produce a contradiction for n ≡ 4,6 mod 8. This is a standard, non-circular application of Farkas' lemma to a fixed linear system; the certificates are constructed algebraically from the constraint matrix without fitting, renaming, or self-referential definitions. No load-bearing self-citation or ansatz is invoked for the infeasibility result. The modeling assumption that the LP relaxation is valid for whirling tours is an external modeling choice, not a circular step inside the proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument relies only on the standard Farkas lemma from linear programming and the combinatorial definition of the directed knight graph; no free parameters or new entities are introduced.

axioms (1)
  • standard math Farkas lemma: a system Ax = b, x ≥ 0 has no solution iff there exists y such that A^T y ≥ 0 and b^T y < 0.
    Invoked to certify infeasibility of the cycle-cover LP relaxation.

pith-pipeline@v0.9.0 · 5418 in / 1235 out tokens · 46144 ms · 2026-05-12T04:11:30.992371+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    E. W. Bennett. Whirling knight’s tours.Fairy Chess Review6, 105 (February 1947), 72, problem 7159; 106 (April 1947), 82

    work page 1947

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    work page arXiv 2021

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    N. I. Beluhov. A proof of Willcocks’s conjecture.arXiv preprint, 2017. arXiv:1708.05810

    work page arXiv 2017

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    D. E. Knuth.The Art of Computer Programming, Pre-Fascicle 8a: A Draft of Section 7.2.2.4 (Hamiltonian Paths and Cycles). Draft of 10 April 2026. Available athttps://cs.stanford. edu/~knuth/fasc8a.pdf

    work page 2026

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    Schrijver.Theory of Linear and Integer Programming

    A. Schrijver.Theory of Linear and Integer Programming. Wiley-Interscience, 1986. 9

    work page 1986