LogFAQs > #962350319

professor_jack posted...

So I can help with this for now.

First just by diagonal and toroidal R2C2(blue) can see all of red:
https://i.imgur.com/Xp1XnZc.png

And just by diagonal and toroidal R2C5(blue) can see all of red:
https://i.imgur.com/umBwMtn.png

As you can see, for the corner center cells and side center cells the diagonal plus normal sudoku rules make it so no center cell can contain the same digit as any other center cell.

Proving this for the center cell is trivial and does not require toroidal rules.

Thus:
Step 1: All center cells are different digits and can be colored differently.

Thanks, yeah these are accurate.

What can help to visualize how toroidal rules work is imagining the grid cloned 8 times around the grid itself. Because that's effectively what happens, if you go off the right side you end up on the left, same for top/bottom and vice versa. So using this hopefully it's more clear why any center cell (blue) can see any others: (sees red from diagonal the other center cells because they are in the same row/column)
https://gamefaqs.gamespot.com/a/user_image/3/7/0/AACZqoAAC4qS.jpg

(Grey grid is just clones of the main grid to help visualize toroidal rules).

So once you know this you can go back to this initial coloring:
https://gamefaqs.gamespot.com/a/user_image/9/5/3/AACZqoAAC135.png

Each color represents 1 of the 9 possible digits.

Next:
https://gamefaqs.gamespot.com/a/user_image/3/7/1/AACZqoAAC4qT.png
So step 2 is to consider any given center cell and completely color everywhere in the grid that cell could be, let's consider the very center cell, that digit (R5C5 purple) could not appear anywhere in yellow because of the bishop's rule, it could not appear anywhere in green because of the knight's move rule, and could not appear anywhere in light blue due to normal sudoku rules.

Thus the only possible location the digit in R5C5 could appear in is without the purple highlighted parts.

Now there are 2 ways forward from here, the way I originally planned it requires you noticing:
https://gamefaqs.gamespot.com/a/user_image/3/7/2/AACZqoAAC4qU.png

Because the digit in R5C5 can only appear in purple, that means it cannot appear in the blue ring above. And I think several sudoku people here like serious_cat could talk about the significance of that ring.

(I'll pause here for today to give people a chance to see if they can get it from here, otherwise I'll finish revealing it over the weekend)