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The basic question is - which one is bigger? This possibly needs some clarification:
By a sudoku puzzle I mean a grid with some cells filled with numbers and others empty so that they can be filled to a complete, valid sudoku grid in a unique fashion. We consider two puzzles the same if they have the same solutions and blanks in the same places. The obvious upper bound is $10^{81}$ as there are $81$ cells which can be either empty or be the number from $1$ to $9$. This can be easily improved to $2^{81} \cdot 6,671,248,172,291,458,990,080 \approx 6 \cdot 10^{46}$ - the number of subsets of all valid sudoku grids. $2^{81} = 2417851639229258349412352$ can be replaced by: $${81 \choose 17} + {81 \choose 18} + \dots + {81 \choose 81} = 2417851595207450142980773$$ As we know that a sudoku must have at least $17$ clues to have a unique solution. As the difference is only minor the estimate doesn't change.
Number of chess positions seems to be more tricky. The upper bound seems to be something around $10^{52}$ (at least according to https://en.wikipedia.org/wiki/Shannon_number) so it might be possible that the number of chess positions is bigger. We will assume that the positions are the same if all their pieces are in the same places, the special move privileges are the same and there is the same player to move. We will ignore $50$ move rule counter and threefold repetitions as it would ramp up this number massively ($50$ move rule only by about $50$ but for the latter we should count for every position so far how many times it has occured).
Is there some proven lower bound that will show that the number of chess positions is bigger than $6 \cdot 10^{46}$?

Bartek
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    Exact number of Sudoku puzzles: https://math.stackexchange.com/questions/275422/how-many-different-sudoku-puzzles-are-there Not sure about chess. – Adam Rubinson Feb 27 '21 at 18:30
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    @Adam Rubinson but this is referring to the number of sudoku grids. I’m talking about number of different puzzles I.e. allowing for blanks in the grid to be filed later. So for example if one puzzle has one more digit placed than the other then we consider them as different, even though they lead to the same solution. – Bartek Feb 28 '21 at 01:23
  • Newer upper bound on chess positions seems to be approx 10^46.7 by John Tromp (from the wiki page you linked). If you follow the reference, at the bottom they claim a new upper bound of 10^45.888, so I do not think you will find a lower bound of 10^46... – Vepir Apr 30 '21 at 20:20
  • On the other hand, a comment on How many different “tight” sudoku puzzles are there? points to wiki Mathematics of Sudoku where there is an estimation that there are only 3.10 × 10^37 (with 0.065% relative error) sudoku puzzles with a unique solution. – Vepir May 02 '21 at 11:01
  • By your own definition, a sudoku puzzle can’t have the same digit twice in a row, column or box, which alone dramatically reduces the number. – gnasher729 Sep 03 '22 at 19:25

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