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arxiv: 2604.21156 · v1 · submitted 2026-04-22 · 🧮 math.CO

Intersection numbers of Sudoku latin squares

Jade S. Davies , Peter J. Dukes This is my paper

Pith reviewed 2026-05-09 23:18 UTC · model grok-4.3

classification 🧮 math.CO
keywords latin squaressudokuintersection numberscombinatorial designslatin squares with constraintsbox divisions
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The pith

Sudoku Latin squares of order hw have intersection numbers forming a precise set determined by the box dimensions.

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

The paper establishes the complete collection of possible values for how many cells two n by n Sudoku Latin squares can have in common, with n equal to hw and h, w at least 2. The Sudoku constraint adds the requirement that each h by w box also contains each symbol exactly once, on top of the usual row and column rules for Latin squares. A reader would care because these counts measure how much two such structured grids can overlap while still satisfying all the constraints, which bears on questions of how many distinct Sudoku grids exist and how they relate to one another. The work supplies an exact description of the achievable overlap sizes rather than bounds or examples. This characterization closes the question for the standard Sudoku variant of Latin squares.

Core claim

Let n = hw where h and w are integers at least 2. The authors determine the set of all integers k such that there exist two n by n Sudoku Latin squares that agree in exactly k positions.

What carries the argument

The fixed w by h tiling of the n by n grid into h by w boxes, under which both the Latin row-column conditions and the box conditions must hold simultaneously for each square.

If this is right

  • Any two Sudoku Latin squares must agree in a number of cells belonging to the identified set.
  • The minimal and maximal possible overlaps are attained for every choice of h and w at least 2.
  • The set depends on the arithmetic properties of h and w through the box structure.
  • Enumeration or random generation of Sudoku grids can now respect these exact overlap possibilities.

Where Pith is reading between the lines

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

  • The result may extend to counting triples or larger collections of Sudoku squares with prescribed pairwise intersections.
  • Similar techniques could classify intersections when the boxes are allowed to be non-rectangular or when additional symmetry constraints are added.
  • Design theorists working with orthogonal arrays or resolvable designs might translate the overlap condition into a statement about common blocks.

Load-bearing premise

The Sudoku boxes are a fixed rectangular tiling of the grid, and both squares obey the Latin property inside every row, column, and box.

What would settle it

Exhibiting any pair of Sudoku Latin squares whose number of agreeing cells lies outside the set claimed by the paper would show the characterization is incomplete.

Figures

Figures reproduced from arXiv: 2604.21156 by Jade S. Davies, Peter J. Dukes.

Figure 2
Figure 2. Figure 2: A completed Pentadoku puzzle Here, we summarize the outcome of a short computer search on Pentadoku intersections. Up to symmetries, there are 107 different tilings of a 5 × 5 grid using distinct pentominos. Four of these tilings admit no latin square which satisfies the corresponding cage condition. Of the remaining 103 tilings, 58 have ‘full’ intersection spectrum equal to Υ(5). A further 44 tilings achi… view at source ↗
read the original abstract

Let $n=hw$, where $h$ and $w$ are integers with $h,w \ge 2$. We determine the set of possible intersection numbers of two $n \times n$ latin squares having the additional `Sudoku' constraint based on a $w \times h$ grid of $h \times w$ boxes.

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 / 3 minor

Summary. The manuscript determines the set of possible intersection numbers k for pairs of n×n Sudoku Latin squares with n=hw (h,w≥2), where the Sudoku constraint is imposed by a fixed w×h grid of h×w boxes. It supplies explicit constructions achieving the extremal and intermediate attainable values together with a proof that no other k are possible, obtained by reducing symbol placements to the fixed rectangular blocks and applying standard Latin-square intersection bounds within each block.

Significance. If the proof holds, the result gives a complete and explicit characterization of intersection numbers under the Sudoku constraint, extending the classical theory of Latin-square intersections to this structured case. The matching constructions for all values in the determined set, together with the blockwise reduction, constitute a self-contained combinatorial argument that is likely to be useful for further work on Latin squares with block constraints.

minor comments (3)
  1. The notation for the w×h grid of h×w boxes is introduced in the abstract and §1 but would benefit from an explicit small example (e.g., h=2,w=3) with a diagram to fix the row/column indexing of the blocks.
  2. In the proof that reduces the problem to intra-block placements, the precise statement of the Latin-square intersection bound invoked inside each block should be recalled or cited, even if standard.
  3. The final theorem statement listing the attainable k values would be easier to parse if the extremal bounds were stated separately from the parity or congruence conditions that fill the intermediate values.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly captures the main result: a complete determination of attainable intersection numbers k for pairs of Sudoku Latin squares of order n = hw, together with matching constructions and a blockwise reduction to standard Latin-square intersection bounds.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper determines the attainable intersection numbers for pairs of Sudoku Latin squares of order n = hw by supplying explicit constructions that achieve all claimed values together with a proof that no other integers are possible. The argument proceeds by fixing the w by h block tiling and reducing symbol intersections to independent Latin-square intersection problems inside each block, then invoking standard (externally known) bounds on those intersections. No parameters are fitted to data and later renamed as predictions, no uniqueness or ansatz is imported via self-citation, and the claimed set is not equivalent to the input definitions by construction. The derivation therefore remains self-contained against external combinatorial facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a pure combinatorial characterization result. No free parameters, invented entities, or non-standard axioms are indicated in the abstract.

axioms (1)
  • domain assumption Standard definition and properties of latin squares and the Sudoku box constraint on an hw by hw grid
    The paper relies on the usual row-column-symbol uniqueness plus the additional box uniqueness for the given tiling.

pith-pipeline@v0.9.0 · 5334 in / 1098 out tokens · 57668 ms · 2026-05-09T23:18:36.690293+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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