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Someone I find very trustworthy continues to insist that this is a valid Sudoku board. It's been a while since I've played, but surely there's a couple of things off here, right?

enter image description here

Alright, so that someone is me. And as it turns out, there are a few things off here. What's the solution to this board? And how does this board work?

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  • $\begingroup$ It's been a while since I've looked at a sudoku. Are the small numbers in a cell nominally the possible values that remain for a given location? Obviously, they aren't here - I just want to make sure I understand what they're playing off of. $\endgroup$
    – Bobson
    Commented Jun 4, 2018 at 21:47
  • $\begingroup$ @Bobson No, the small numbers are not related to penciling in values at all $\endgroup$
    – JMigst
    Commented Jun 4, 2018 at 21:55
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    $\begingroup$ I think your puzzle might be a little too enigmatic. I see no information that really gives the solver a place to start. :( $\endgroup$
    – dcfyj
    Commented Jun 29, 2018 at 12:51
  • $\begingroup$ @dcfyj Well among other things, the small numbers represent another puzzle and connect to the fractional values. $\endgroup$
    – JMigst
    Commented Jun 29, 2018 at 14:13
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    $\begingroup$ @LuisSousa I'm not sure if the OP is still around but I believe it is. I'm guessing it is similar to my Unusual Sudoku Puzzle where the last step is to solve the Sudoku normally. $\endgroup$
    – Bennett Bernardoni
    Commented Aug 29, 2018 at 14:34

2 Answers 2

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Here is the final solution.

Step 1

The small numbers reminded me of a nonogram and sure enough it has the following solution. 

        12  3
       3111323
      +-------+
    2 |    XX |
  1 2 |  X XX |
  2 2 | XX XX |
      |       |
  2 1 |XX    X|
  1 2 |X    XX|
1 2 2 |X XX XX|
      +-------+
In the comments, the author hinted that the solution to the nonogram connects to the fractional values. So, cross checking with the original puzzle we find that all of the fractional values line up with the Xs. Furthermore, the denominator of the fraction is related to the amount of contiguous Xs. This means we can multiply by the number of contiguous Xs to get integers. The numbers on the board now looks like the picture below, where the nonogram squares are higlighted.

Sudoku solution 1

Step 2

The board still has invalid numbers, so these must be corrected with the chess pieces. At first I thought the pieces should move like normal on a 9x9 board. However, as hinted in the comments the chess pieces behave differently. Instead they move on the 3x3 grid of larger boxes while keeping their position in the box. For example the -3 king can move to the 10 in the first box. After the chess pieces make their move, their number is added to the cell. In the previous example the 10 would become a 7. Doing this to all the chess pieces results in the following image with the changed cells highlighted.

Sudoku solution 2

Step 3 and solution

Finally, we have a valid Sudoku board and just need to solve it normally. This gives the following solution.

Sudoku solution 3

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  • $\begingroup$ 10 isn't reachable only if the chess moves are small (the board is bigger than you think). $\endgroup$
    – JMigst
    Commented Aug 29, 2018 at 15:03
  • $\begingroup$ @JMigst I meant only reachable in one move. I figured that the pieces can move outside their 3x3 houses (which is maybe what you meant by the board is bigger than I think) but this doesn't help reaching the 10 in one move. Although multiple moves may be necessary, I guess. $\endgroup$
    – Bennett Bernardoni
    Commented Aug 29, 2018 at 16:36
  • $\begingroup$ That's still not what I meant! You only need one move per chess piece. In hindsight, the behaviour of the chess pieces are probably the least intuitive part, but hopefully it's clear once you get it. $\endgroup$
    – JMigst
    Commented Aug 29, 2018 at 21:52
  • $\begingroup$ @JMigst Thanks, for the hint. After realizing that the chess pieces behaved differently, I figured it out. $\endgroup$
    – Bennett Bernardoni
    Commented Aug 29, 2018 at 23:39
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I tried doing some analysis of the chess pieces. I didn't find anything useful, but I figured I'd share what I had in case it helps anyone else:

Board positions (assuming standard locations extend an extra square) and the number on each piece:

Queen  a3 (2)
Bishop c8 (3)
Rook   e1 (1)
Knight e3 (5)
Bishop e4 (8)
Rook   e6 (7)
King   f9 (4)
Queen  h4 (3)

Knight b5 (-3)
King   e8 (-3)

I also noted that the sequence of all the numbers is -3, -3, 1, 2, 3, 3, 4, 5, 7, 8. If you remove the 3's and -3's (they cancel out?) you get 1, 2, _, 4, 5, _, 7, 8. That's repetitive enough that it might lead to something, but I don't see anything useful to come of it.

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  • $\begingroup$ There is two of every chess piece, I noticed. $\endgroup$
    – Mr Pie
    Commented Aug 1, 2018 at 23:36
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    $\begingroup$ @user477343 - Huh. I missed that. Interesting. No idea what to make of it, but interesting. $\endgroup$
    – Bobson
    Commented Aug 2, 2018 at 3:51

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