Sudoku Grids That Require Many Clues

Abstract: Motivated by worst-case algorithmic time bounds for solving sudoku, we prove that a majority of filled-in $n^2\times n<sup>2$ sudoku grids require all but a logarithmic fraction of cells to be filled by clues. For $9\times 9$ and $16\times 16$ sudoku, we construct grids that require $18$ clues and $80$ clues.

• The paper proves that almost all filled n²×n² Sudoku grids require n⁴−O(n⁴/log n) clues, establishing that sparsely specified grids are exceptionally rare.
• The paper develops an inclusion–exclusion solver running in n^{O(1)}2^{n⁴−m} time, yielding a 2^{O(n⁴/log n)} average-case bound for nearly all grids and minimal clue sets.
• The paper constructs grids requiring at least 18 clues for 9×9 Sudoku and 80 clues for 16×16 Sudoku by packing Latin squares, substantially exceeding the conjectured 56-clue bound for 16×16 puzzles.

Overview and problem statement

This paper, by Eppstein and Zhang, studies a worst-case variant of the minimum-clue problem for sudoku. Rather than asking how few clues suffice for some puzzle — the classical question, whose answer is 17 for $9\times 9$ sudoku — the authors ask: given a filled-in $n^2\times n^2$ sudoku grid, what is the minimum number of clues that uniquely determine it as a solution? They then seek grids that maximize this quantity. The motivation is algorithmic: the fastest known general-purpose sudoku solver runs in time exponential in the number of empty cells, so grids that force nearly all cells to be clued are precisely the hard instances for such methods.

The paper's two main contributions are a counting argument showing that almost all filled grids require $n^4 - O(n^4/\log n)$ clues — i.e., all but a logarithmic fraction of cells must be given as clues — and explicit constructions of grids requiring 18 clues ($9\times 9$) and 80 clues ($16\times 16$). The latter figure substantially exceeds the conjectured minimum of 56 clues for $16\times 16$ sudoku (2607.05728).

Algorithmic motivation and solver

Sudoku is NP-hard (2607.05728), yet published puzzles are typically solvable by polynomial-time deduction rules. To narrow the gap between theory and practice, the authors adapt the inclusion–exclusion framework of Björklund, Husfeldt, and Koivisto for exact cover to the "sum weighted partitions" problem. A valid placement of digit $i$ is a set of cells containing one cell in each row, column, and block; a solution partitions all cells into $n^2$ valid placements, one per digit. Setting $f_i(S)=1$ iff $S$ is a valid placement of digit $n^2\times n^2$0, the sum weighted partitions value equals the number of solutions (the paper does not assume this value is one), which can be evaluated to decide solvability and, by iteration over digit placements, to solve the puzzle.

Theorem (solver bound). Sudoku with $n^2\times n^2$1 clues is solvable in time $n^2\times n^2$2.

Valid placements are enumerated via a reduction to path enumeration in graphs. The consequence is direct: since runtime scales exponentially in unfilled cells, grids requiring many clues are exactly the instances on which this method performs best, which motivates quantifying how large the required clue count can be.

Existence via counting

Let $n^2\times n^2$3 be the number of filled $n^2\times n^2$4 grids. Only $n^2\times n^2$5 and $n^2\times n^2$6 are known exactly; asymptotically, Keevash's result gives $n^2\times n^2$7. The number of ways to specify $n^2\times n^2$8 clues satisfies $n^2\times n^2$9, counting clue positions and digit assignments independently. Since each choice of $n^4 - O(n^4/\log n)$0 clues determines at most one filled grid, and puzzles with fewer clues can be extended to exactly $n^4 - O(n^4/\log n)$1 without changing the count, comparing $n^4 - O(n^4/\log n)$2 against $n^4 - O(n^4/\log n)$3 yields:

Theorem (almost-all lower bound). All but a $n^4 - O(n^4/\log n)$4 fraction of filled $n^4 - O(n^4/\log n)$5 sudoku grids require $n^4 - O(n^4/\log n)$6 clues.

The proof takes base-2 logarithms of both sides, uses $n^4 - O(n^4/\log n)$7, and consolidates lower-order terms. Because every filled grid has at least one minimal clue set, the same counting argument applies to minimal clue sets themselves. This immediately improves the exponent of the exponential-time solver:

Corollary. For all but a $n^4 - O(n^4/\log n)$8 fraction of filled grids or minimal clue sets, solving takes time $n^4 - O(n^4/\log n)$9, and the same bound holds in the average case over random grids or minimal clue sets.

Thus the worst-case $9\times 9$0-type behavior of the inclusion–exclusion solver is avoided on essentially all inputs, including random ones. The implication is notable: average-case sudoku solving is exponentially faster than the naive worst-case bound suggests, by a factor exponential in $9\times 9$1.

Explicit constructions from packed Latin squares

The constructive contribution packs $9\times 9$2 small Latin squares into an $9\times 9$3 sudoku grid. The procedure has three steps: partition the $9\times 9$4 digits into $9\times 9$5 subsets $9\times 9$6 of size $9\times 9$7; use an $9\times 9$8 Latin square per horizontal group of blocks to assign digit subsets to block rows; and, within each vertical group of blocks, fill each subset's assigned rows as an $9\times 9$9 Latin square. The number of distinct grids produced is $16\times 16$0, where $16\times 16$1 counts $16\times 16$2 Latin squares.

For $16\times 16$3, since $16\times 16$4, this yields $16\times 16$5 distinct $16\times 16$6 grids. Each embedded $16\times 16$7 Latin square requires at least two clues — a single clue would permit swapping the other two digits within that square — so each grid requires at least $16\times 16$8 clues. This matches McGuire et al.'s computational finding that 17 clues do not always suffice, but here it is established constructively rather than by exhaustive search.

For $16\times 16$9, case analysis shows that the Klein-type Latin square

$16\times 16$0

requires five clues. Packing 16 copies into a $16\times 16$1 grid produces a grid requiring 80 clues, well above the conjectured 56-clue minimum for $16\times 16$2 sudoku. This is the paper's most striking numerical claim: it demonstrates that the conjectured minimum, if correct, applies only to some grids, not to typical or worst-case ones.

In general, Hatami and Qian prove that every $16\times 16$3 Latin square requires $16\times 16$4 clues, so the construction yields sudoku grids requiring $16\times 16$5 clues for all $16\times 16$6. This aligns the construction with the conjectured $16\times 16$7 minimum-clue growth rate, though the conjecture itself remains open — only $16\times 16$8 is proven unconditionally for arbitrary grids.

Limitations and open questions

Several caveats bear directly on the results. The almost-all theorem is existential: it shows most grids are hard to specify sparsely but identifies no explicit hard grid beyond the constructed families, and its constant hidden in the $16\times 16$9 term is not optimized. The construction's lower bounds rely on the local argument that each embedded Latin square needs at least two (or five) clues; the paper does not verify that these bounds are tight, i.e., that 18 and 80 clues actually suffice for the constructed grids. The authors also leave open the computational complexity of computing the minimum number of clues for a given filled grid — noting only that the related completion problem is $i$0-hard — and do not address whether their counting-based average-case speedup extends to other sudoku algorithms such as backtracking.

Conclusion

The paper reframes the minimum-clue problem around worst-case filled grids and delivers both a strong probabilistic result — almost all grids require $i$1 clues, yielding $i$2 average-case solving time for inclusion–exclusion methods — and concrete witnesses at sizes 9 and 16, including a $i$3 grid needing 80 clues. Together these results show that sparse uniquely-solvable puzzles are exceptional objects, while typical grids are information-theoretically dense, and they sharpen the known landscape between the proven $i$4 and conjectured $i$5 lower bounds for generalized sudoku.