There are only twelve 4x4 sudokus - and a cool trick for finding minimal subsets

There are only twelve unique 4x4 sudoku solutions, but there are 288 possible solutions when considering distinct structures. These solutions are derived from a 4x4 grid divided into rows, columns, and 2x2 boxes, with the goal of filling each cell with a digit from 1 to 4, ensuring that each digit appears exactly once in every row, column, and box. This is a smaller case of the standard 9x9 sudoku puzzle.

A brute force approach can be used to solve this question, as the number of possibilities is quite small. By starting with an empty grid and recursively exploring each possible value for each cell, it can be determined that there are 288 possible 4x4 sudoku solutions.

However, some of these solutions are structurally identical, as swapping digits doesn't change the underlying structure. For instance, if solution (A) is a valid solution, swapping all 1s with 2s and vice versa results in a new solution (B) that is structurally identical to (A). This observation allows us to understand that there are only twelve distinct 4x4 sudoku structures.

When considering the number of distinct 4x4 sudoku puzzles with a minimal number of given digits, the problem becomes more complex. To find the answer, a brute force approach can be employed, which involves initializing an empty sudoku grid with the starting configuration 1 2 3 4 in the first row. This approach confirms the previously calculated result of twelve distinct structures, and provides a way to count the number of minimal sudoku puzzles.