In how many ways can we fill a $9\times 9$ matrix with digits from 1 to 9 so that all rows, all columns and all the nine $3\times3$ submatrices [obtained after partitioning the bigger matrix into nine $3\times3$ matrices] contain all the digits from 1 to 9? In other words, what is total number of solutions of a completely blank sudoku. I have been grappling with this problem for months without success. Any help is greatly appreciated.
If you type "total number of sudoku solutions" into Google then the first hit is Wikipedia's Sudoku article. Within it is a subsection called "Enumerating the Sudoku 9×9 grid solutions directly". The answer seems to be 6,670,903,752,021,072,936,960.
You might like to read this research article.