Snipes

This page is for collecting "nerd snipes" - various tasks to answer questions about minimums, maximums, and counts for all things sudoku.

Google Doc - this page needs to be populated with information from here.

Classic Sudoku

The fewest givens in a classic sudoku is 17; this was proven by McGuire et al. There exist 16 given sudoku with as few as 2 solutions. There are exactly 49158 essentially different 17 given puzzles; the last was found in 2019, and an exhaustive search (link) has confirmed that this is a complete list. List of all essentially different 17 given puzzles.

Every solution grid has a minimal puzzle with fewer than 21 givens. There are exactly two grids which do not have a minimal puzzle with fewer than 20 givens. Template:Citation needed

For 6x6 (standard, with 3x2 regions), the minimum is 8 givens (link).

Irregular Sudoku

The fewest givens with irregular regions is 8. This is the theoretical minimum, as at least this many digits are needed to disambiguate the digit permutation.

Variants

Unless otherwise noted, the minimums for variants are given either for the fewest givens needed for a unique solution (for a global variant), the fewest of a constraint needed for a unique solution with no given digits, or the fewest cells covered by a constraint needed for a unique solution with no given digits.

Anti-Knight

The fewest givens in an anti-knight sudoku is 8; there are 4 essentially different puzzles, found initially by ryokousha, rangsk, and PN. An exhaustive search using a Gauntlet showed that these are the only four (link). This is the theoretical minimum, as at least this many digits are needed to disambiguate the digit permutation.

If givens are restricted to the borders of the grid, the fewest possible is 10, found by PN (link). For 9 givens, the lowest solution count is 18, found by PN and ryokousha; this has been verified by rangsk.

Anti-King

The fewest givens in an anti-king sudoku is 11, found by Philip Newman in collaboration with ryokousha (link). It is believed this is minimal, but this has not been proven.

Anti-King + Anti-King

With both anti-king and anti-knight constraints, there is exactly one “essentially different” solution grid (which can be rotated, reflected, or have the digits permuted, to obtain the full list of possible solutions). The fewest possible givens remains 8 in order to disambiguate the digit permutation.

Diagonal (Sudoku X)

The fewest givens known in a diagonal sudoku is 12; over 500,000 such puzzles are known Template:Citation needed. It is not known whether 11 is possible, though a number of 11c puzzles with 2 solutions are known (including four with 8 distinct digits, found by ryokousha).

Diagonal + Anti-King

The fewest givens in a sudoku with both diagonal and anti-king constraints is 9, found by PN (link); there are 3 examples known currently. This is believed to be minimal, but this may be proven or disproven by exhaustive search (see above Anti-Knight section).

Arrow

The fewest arrows is 5, and the fewest cells covered is 21, both found by PN in collaboration with (based on a puzzle by) BlueJay (link). It is unknown whether either is minimal.

Arrow + Anti-Knight

The fewest arrows (single digit sums) is 3, found by PN (link). This is the theoretical minimum (with two arrows you can have at most seven distinct digits).

If two-digit sums are allowed then 1 arrow is sufficient, as found by Gliperal (link).