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.
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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.