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How many different sudoku puzzles are there? posted a similar question and Chris Eagle's answer https://math.stackexchange.com/a/275425/26632 points to http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf saying that there are $6,670,903,752, 021,072,936,960$ different grids.

However, grid is not puzzle:

  • a grid is a final status, with all 81 cells known; where a puzzle is the initial status, some difficult sudoku puzzles starts with only 17 or 18 cells given.
  • for a given grid, masking different cells will start a different puzzle.
  • a puzzle shall have but only 1 solution, ie. 1 grid. There shall be no ambiguity, i.e. a puzzle shall not have multiple solutions.
  • to be neat, we prefer to start with a tight puzzle, i.e., the puzzle provides necessary information but no more, or, masking any of the numbers provided will lead to ambiguity.

So, how many sudoku tight puzzles are there? As explained above, $6,670,903,752, 021,072,936,960$ is the answer only for grid but not for puzzle count.

If this is too difficult, how many $4\times 4$ sudoku tight puzzles are there?

Rodrigo de Azevedo
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athos
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    According to the article referenced at Wikipedia: For $9\times 9$ this long-standing mathematical problem has not been solved, but statistical simulations (using several months of $3$ GHz CPU time) indicate that on average about $4.67,10^{15}$ tight puzzles can be made from each grid; in other words, there are about $3.10,10^{37}$ different tight puzzles, and about $2.55,10^{25}$ tight puzzles that are non-isomorphic. – r.e.s. May 22 '20 at 14:48

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