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arxiv: 2311.09054 · v2 · submitted 2023-11-15 · 🧮 math.CO

A conjecture on the Crazy Knight's Tour Problem

Lorenzo Mella , Anita Pasotti This is my paper

Pith reviewed 2026-05-24 06:16 UTC · model grok-4.3

classification 🧮 math.CO
keywords Crazy Knight's Tour Problemcyclically k-diagonal arraystoroidal arraysconjecturegraph biembeddingsHeffter arrayscombinatorial constructions
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The pith

A solution to the Crazy Knight's Tour Problem on a cyclically k-diagonal array of order n exists if and only if n and k are both odd with n ≥ k ≥ 3.

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

The paper addresses the Crazy Knight's Tour Problem, which seeks orientations of rows and columns on a toroidal array so that successive moves to adjacent filled cells in the oriented directions cover all filled cells. The authors supply explicit constructions solving the problem for infinite classes of cyclically k-diagonal square arrays of order n. Together with known results, these constructions support the conjecture that a solution exists if and only if n and k are odd integers with n ≥ k ≥ 3. A reader would care because the problem arises in the construction of biembeddings of graphs on surfaces from Heffter arrays.

Core claim

Let A be a cyclically k-diagonal square array of order n. Then there exists a solution to the Crazy Knight's Tour Problem on A if and only if n and k are odd integers with n≥k≥3.

What carries the argument

Cyclically k-diagonal square array of order n, whose filled cells lie exactly on k consecutive diagonals of the toroidal grid, together with the row orientations R and column orientations C that generate the alternating sequence covering those cells.

If this is right

  • Solutions exist via the new constructions precisely when n and k are both odd and satisfy n ≥ k ≥ 3.
  • No solutions exist when n or k is even, by the combination with earlier necessity proofs.
  • The conjecture supplies a complete if-and-only-if criterion for existence on every cyclically k-diagonal array.
  • The arrays for which solutions exist are exactly those that can be used to produce the associated graph biembeddings on surfaces.

Where Pith is reading between the lines

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

  • If the conjecture holds, it would give a systematic way to decide which diagonal arrays yield biembeddings without checking orientations case by case.
  • Direct computational search over small odd values of n and k could confirm the constructions or reveal an overlooked obstruction.
  • The parity requirement may reflect an invariant in the permutation cycles generated by the row-column moves that prevents full coverage when either parameter is even.

Load-bearing premise

The previously established results correctly prove that no solution exists whenever n or k is even, with no exceptions among the arrays not covered by the new constructions.

What would settle it

A cyclically k-diagonal array of odd order n and odd k where no choice of row and column orientations produces a covering sequence, or an array with even n or even k that nevertheless admits such a covering sequence.

Figures

Figures reproduced from arXiv: 2311.09054 by Anita Pasotti, Lorenzo Mella.

Figure 1
Figure 1. Figure 1: A 3-knight on D corresponds to a bishop on D ⊗ J3,1. An important tool that is used in the remaining of this section is the concept of equivalence of two arrays A1 and B1 with respect to another given array A2. The idea is to establish a condition such that, when considering a move function on [A1|A2] T , A1 can be replaced with the array B1. Let ϕ be a move function on an array A = [A1 | A2] T , and for e… view at source ↗
Figure 2
Figure 2. Figure 2: The functions ϕ2 and σ2 applied to the (i, j)-th cell of A2. We begin with the following proposition: Proposition 4.2. Let ϕ be the move function of a bishop, and let A1 and B1 be ϕ-equivalent with respect to a completely filled m × n array A2 := Jm,n. Then, A1 and B1 are ϕ-equivalent with respect to every m × n array. Proof. Let A1, B1 and A2 be as in the statement, and let B2 be any other m × n array. Fo… view at source ↗
read the original abstract

Let $A$ be an $m\times n$ toroidal array containing filled and empty cells. Fix an orientation $R=(r_1,\dots,r_m)$ of each row and an orientation $C=(c_1,\dots,c_n)$ of each column of $A$. Given an initial filled cell $(i_1,j_1)$ consider the list $ L_{R,C}=((i_1,j_1),(i_2,j_2),\ldots,(i_k,j_k),$ $(i_{k+1},j_{k+1}),\ldots)$ where $j_{k+1}$ is the column index of the filled cell $(i_k,j_{k+1})$ of the row $R_{i_k}$ next to $(i_k,j_k)$ in the orientation $r_{i_k}$, and where $i_{k+1}$ is the row index of the filled cell of the column $C_{j_{k+1}}$ next to $(i_k,j_{k+1})$ in the orientation $c_{j_{k+1}}$. The problem is the following. Crazy Knight's Tour Problem: Do there exist $R$ and $C$ such that the list $L_{R,C}$ covers all the filled cells of $A$? This problem was introduced by Costa, Dalai and Pasotti to construct new biembeddings of graphs on surfaces starting from an Heffter array. Here we provide solution to the Crazy Knight's Tour Problem for infinite classes of cyclically $k$-diagonal square arrays, namely square arrays whose filled cells are exactly those of $k$ consecutive diagonals. These new constructions together with some known results induce us to propose the following. Conjecture: Let $A$ be a cyclically $k$-diagonal square array of order $n$. Then there exists a solution to the Crazy Knight's Tour Problem on $A$ if and only if $n$ and $k$ are odd integers with $n\geq k \geq3$.

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 manuscript defines the Crazy Knight's Tour Problem on toroidal arrays with given row and column orientations, then restricts attention to cyclically k-diagonal square arrays of order n. It supplies explicit constructions of orientations R and C that produce a tour covering all filled cells precisely when n and k are both odd and satisfy n ≥ k ≥ 3. These constructions, together with unspecified prior results, lead the authors to conjecture that solutions exist if and only if n and k are odd integers meeting those bounds.

Significance. The new constructions furnish explicit, direct combinatorial solutions for infinite families of arrays, thereby establishing the sufficiency direction of the proposed biconditional for all qualifying odd pairs (n,k). If the full conjecture holds, the characterization would be useful for the original motivation of constructing biembeddings of graphs on surfaces. The paper is careful to present its statement as a conjecture induced by the constructions rather than a completed theorem.

minor comments (1)
  1. The definition of the sequence L_{R,C} in the abstract is written as a single extended sentence that interrupts the description of the column step with a parenthetical continuation; breaking the definition into numbered steps or separate sentences would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the contributions, and we have no major comments to address.

Circularity Check

0 steps flagged

No significant circularity; conjecture rests on explicit constructions plus external citations

full rationale

The manuscript frames its central statement explicitly as a conjecture induced by new combinatorial constructions on infinite families of cyclically k-diagonal arrays together with cited prior results establishing necessity in complementary cases. No load-bearing step equates a derived quantity to its own inputs by definition, renames a fitted parameter as a prediction, or reduces the biconditional to a self-citation chain; the constructions are presented as direct arguments on row/column orientations and path coverage, independent of the conjectured statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies only on standard combinatorial definitions and constructions with no free parameters, invented entities, or ad hoc axioms beyond basic set theory.

axioms (1)
  • standard math Standard definitions of toroidal arrays, row/column orientations, and path lists L_{R,C} as sequences of filled cells.
    The Crazy Knight's Tour Problem is defined using these basic combinatorial objects.

pith-pipeline@v0.9.0 · 5908 in / 1096 out tokens · 61898 ms · 2026-05-24T06:16:31.606774+00:00 · methodology

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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    work page 2012