Suirodoku

Mathematical foundations

The concept of superimposing two orthogonal Latin squares to create unique pairs dates back to Leonhard Euler's work in the 18th century on Graeco-Latin squares (or Euler squares).^[2]

The extension of this concept to Sudoku squares (Latin squares with the additional constraint of 3×3 regions) was formalized by J. Subramani and K.N. Ponnuswamy in 2009, who introduced "Sudoku designs" as tools for experimental design in statistics.^[1] In the same year, Pedersen and Vis independently constructed sets of mutually orthogonal Sudoku Latin squares (MOSLS) for any order k², establishing a lower bound on the attainable size of such sets.^[3]

Academic development

In 2012, J. Subramani published the first systematic method for constructing Graeco-Sudoku squares of odd orders, including complete 9×9 grids satisfying all constraints.^[4] This publication demonstrates the existence of valid grids and provides a construction algorithm based on developing initial rows.

In 2013, Subramani further developed the analysis of orthogonal (Graeco) Sudoku square designs, establishing the theoretical framework for these structures.^[5]

Subsequent work extended these concepts:

• 2009: Lorch established mutually orthogonal families of maximal size for linear Sudoku solutions of all square orders.^[6]
• 2017: Hussain et al. introduced the "Hyper Graeco Latin Sudoku Square Design" allowing three sets of treatments to be tested simultaneously.^[7]
• 2017: D'haeseleer et al. proved the existence of non-extendible sets of MOSLS, showing the maximum number of MOSLS of order n=m² is n−m.^[8]
• 2019: Hussain et al. developed parameter estimation under random and mixed effects models.^[9]
• 2023: Kubota, Suda, and Urano introduced graph-theoretic methods to study MOSLS, using eigenvalues to distinguish non-isomorphic Sudoku Latin squares.^[10]
• 2025: Sahib et al. introduced the "Hyper Block Graeco Latin Sudoku Square Design" with row and column blocking to further reduce experimental error.^[11]

Consumer adaptation

The term "Suirodoku" was introduced in 2025–2026 as a consumer-friendly adaptation of the mathematical concept of Graeco-Sudoku squares, replacing the second set of digits with colors for better visual accessibility.^[12] This presentation makes the puzzle more intuitive while preserving the underlying mathematical structure.

Maire (2026) published an overview paper establishing Suirodoku's position within the hierarchy of combinatorial structures, providing a CSP formalization with 55 constraints and proving a bijection theorem.^[13] A companion paper formalized the puzzle within the Multi-Sorted First Order Logic framework established by Berthier.^[14] A third paper introduced the "God Digit Problem" and proved a dichotomy theorem on critical symbols.^[15]

Grid

The Suirodoku grid is identical to that of Sudoku: 81 cells organized into 9 rows, 9 columns, and 9 regions of 3×3. However, each cell contains two pieces of information: a digit (from 1 to 9) and a color (from 9 distinct colors).

Constraints

Suirodoku imposes three categories of constraints:^[14]

1. Row constraint: each row contains digits 1 to 9 without repetition and all 9 colors without repetition.
2. Column constraint: each column contains digits 1 to 9 without repetition and all 9 colors without repetition.
3. Region constraint: each 3×3 region contains digits 1 to 9 without repetition and all 9 colors without repetition.
4. Global pair uniqueness constraint: each of the 81 digit-color pairs (9 digits × 9 colors) appears exactly once in the grid.

The first three are local constraints (each involving 9 cells); the fourth is a global constraint linking all 81 cells simultaneously. In the CSP formalization, this yields 55 constraints total: 27 for digits (9 rows + 9 columns + 9 blocks), 27 for colors, and 1 global pair uniqueness constraint.^[14]

Bijection property

The global pair uniqueness constraint induces a bijection between cells and pairs: the content of each cell uniquely identifies it among all 81 cells. This "absolute cell identity" property is absent from classical Sudoku, where cells containing the same digit are logically interchangeable.^[13]

Formally, the mapping φ from cells to digit-color pairs is well-defined (each cell contains exactly one pair), injective (no two cells share a pair), and surjective (81 cells, 81 pairs). This was proved as the Bijection Theorem in the CSP formalization.^[14]

Techniques inherited from Sudoku

Classic Sudoku techniques apply to Suirodoku, for both digits and colors:

• Naked single
• Hidden single
• Naked and hidden pairs and triples
• X-Wing, Swordfish, XY-Wing

These techniques can be used in parallel on the numerical dimension and the chromatic dimension.

Digit tracking technique ("Rainbow")

This technique exploits the global pair uniqueness constraint. It consists of tracking a given digit across the nine colors:^[12]

1. Identify all occurrences of a digit (e.g., 4) on the grid.
2. Note the color of each occurrence.
3. If eight colors are present, the missing color is determined.
4. Locate the cell where this digit-color pair can be placed.

This approach is analogous to analysis techniques for Graeco-Latin squares in experimental design.^[5]

Color tracking technique ("Chromatic Circle")

Symmetrical to the previous one, it consists of tracking a color across the nine digits:^[12]

1. Identify all cells of a given color (e.g., green).
2. Note the digit of each cell.
3. If eight digits are present, the missing digit is determined.
4. Locate the cell where this pair can be placed.

Both heuristics are consequences of the global pair uniqueness constraint and have no analogue in classical Sudoku.^[13]

Combinatorial complexity

The number of valid classic Sudoku grids is 6,670,903,752,021,072,936,960 (approximately 6.67 × 10²¹).^[16]

For Graeco-Sudoku squares, the number of valid configurations is considerably reduced by cross-constraints. Subramani (2012) demonstrated that for a given Sudoku grid of order m², one can construct m² distinct Graeco-Sudoku squares, and that there exist (m!)^2(m+1) × m² distinct Graeco-Sudoku squares of order m².^[4]

For m = 3 (9×9 grids), this gives:

• 9 distinct Graeco-Sudoku squares for each base Sudoku grid
• (3!)^2×4 × 9 = 6⁸ × 9 = 15,116,544 distinct Graeco-Sudoku squares in total

The maximum number of mutually orthogonal Sudoku Latin squares of order n = m² is n − m, as established by D'haeseleer et al.^[8] For n = 9, this gives a maximum of 6 MOSLS.

The number of essentially distinct Suirodoku grids (equivalence classes under the standard symmetry group) is unknown.^[15]

CSP formalization

Maire (2026) formalized Suirodoku as a Constraint Satisfaction Problem within the Multi-Sorted First Order Logic framework established by Berthier.^[17] The theory comprises six sorts (Number, Row, Column, Block, Square, Color) each of cardinality 9, and ten axioms governing cell uniqueness, completeness, and the local/global constraints. The key axiom ST_global (global pair uniqueness) distinguishes Suirodoku from classical Sudoku and creates a global dependency linking all 81 cells.^[14]

Construction algorithms

Subramani (2012) presented a sequential construction method based on developing two initial rows A₁ and B₁ satisfying certain orthogonality conditions.^[4] This approach is more efficient than exhaustive backtracking search.

Pedersen and Vis (2009) provided an alternative construction using finite fields and coset representatives, generalizable to any square order.^[3]

For applications in experimental design, these constructions allow studying multiple factors simultaneously with optimal variance control.^[7]

Minimum clues

For classical Sudoku, McGuire, Tugemann, and Civario proved that no 16-clue puzzle has a unique solution, establishing 17 as the minimum.^[18] The corresponding minimum for Suirodoku is unknown.

God Digit Problem

Maire (2026) posed the God Digit Problem: must every uniquely solvable Suirodoku puzzle contain all 9 digits among its clues?^[15] In classical Sudoku, any digit can be omitted by relabeling. In Suirodoku, the global pair uniqueness constraint breaks this symmetry. The Dichotomy Theorem proves that exactly one of two outcomes holds: either no digit is critical (for every digit, there exists a uniquely solvable puzzle omitting it), or all digits are critical (every uniquely solvable puzzle must contain all 9 digits). An analogous dichotomy holds independently for colors. Whether digit criticality implies color criticality remains open.^[15]

The name alludes to "God's number" in Rubik's Cube theory.

Unavoidable sets

The God Digit Problem is closely related to the structure of unavoidable set in Suirodoku. In classical Sudoku, for any two digits, the 18 cells containing them form an unavoidable set (swapping preserves validity). In Suirodoku, the global pair uniqueness constraint changes which trades are admissible. Three plausible trade types have been identified: inherited (classical digit swaps preserving color constraints), color-based, and coupled (simultaneous digit-color changes). A systematic classification remains open.^[15]

Other open questions

• Complete enumeration of distinct Suirodoku grids
• Existence of Suirodoku for other orders: 4×4 (2×2 blocks) or 16×16 (4×4 blocks)
• Formal difficulty classification for Suirodoku puzzles