A 9X9 board follows standard Sudoku rules i.e. no two numbers can repeat in the same horizontal row, vertical column or 3x3 sub-square, now 9 numbers 1...9 are randomly scattered throughout the entire board without repetition. Since the numbers are all unique the board is always trivially valid with no need for additional checks, the question now is what percentage of these configurations are solvable. (they may have more than one solution)

Note:-

i) A valid set of clues following Sudoku rules may not always be solvable as it may create dead ends later.

ii) There are 6.67 sextillion(6.67 x 10\^21) completed Sudoku board configurations.

We can also generalise this problem for any N\^2 x N\^2 board following sudoku rules and having N\^2 clues scattered randomly throughout the entire board without repetition, Interestingly for N=2 i.e. a 4x4 board we get 100% solvability this can be verified by solving all possible configurations as there arent that many really about 16!/12!(43680) ways, though it isnt possible to use brute force computation for larger values, I tried to solve this problem using combinatrics but I feel graph theory would have a better answer to this problem.

An interesting problem based on the solvability of a Sudoku puzzle.