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I have to prove that if you put some random number (between 1 and 9) at any of the fields of an empty Sudoku, it is always solvable, but I do not find the way to it.

One way to prove it would be solving the 9*81 possible initial values, but doesn't seem like a proper way to proceed.

I also did think about formulating the problem as an equation system and prove that it has many solutions, but I'm not sure how to do this.

Any idea out there?

Thanks.

Alberto Castaño
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  • Well, it is not true, you could for instance get twice the same number in a column, which would make the sudoku unsolvable. There are probably missing hypothesis here. – servabat Mar 03 '17 at 10:15
  • @servabat As I understood, we only give one cell a value. This can be trivially solved because we can permute the numbers $1-9$ and still have a valid sudoku. – Peter Mar 03 '17 at 10:15
  • @Peter It is possible to renumber a sudoku to place a number wherever you want? – Alberto Castaño Mar 03 '17 at 10:18
  • If you only fix one entry, yes! – Peter Mar 03 '17 at 10:19
  • I meant a solved sudoku. – Alberto Castaño Mar 03 '17 at 10:19
  • So, you're given an empty grid, you fill only one cell with a random number, and you want to show that this grid is solvable? – servabat Mar 03 '17 at 10:20
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    You can take a solved sudoku and permute it that way that one given cell has a given value. For example, you can exchange all the ones and twos and have again a valid sudoku. – Peter Mar 03 '17 at 10:21
  • @servabat That is the idea, yes. you could place any cell whit any number. – Alberto Castaño Mar 03 '17 at 10:21
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    @Peter Well, I think I'll take that as a valid answer since it solved my problem. Never thought about it. Thank you. :) – Alberto Castaño Mar 03 '17 at 10:22
  • @AlbertoCastaño Even better, you can fix a whole column or row or block (of course without a duplicate number) and still can always solve the sudoku. – Peter Mar 03 '17 at 10:46

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