# 9x9 Sudoku Slitherlink Hybrid

I previously created a combined slitherlink and sukoku puzzle, but I was unsatisfied because the grid was smaller then a regular sudoku. So I made another, larger one.

Below is a 9 by 9 grid. This is filled in with numbers based on the usual rules of sudoku. All of the numbers from 1 to 9 must appear in every row, column, and block.

A single, closed loop must also be drawn on the edges using the rules of slitherlink. The numbers describe how many edges adjacent to the square the loop passes through. Because a square only has 4 edges, the loop must use the number modulo 4. Numbers with a red background are liars. Liars are always incorrect, thus the loop must use a different number of edges. Since these rules are complicated, a table can be found below.

1 1 0,2,3
2 2 0,1,3
3 3 0,1,2
4 0 1,2,3
5 1 0,2,3
6 2 0,1,3
7 3 0,1,2
8 0 1,2,3
9 1 0,2,3

And also, the above grid has been converted into a more traditional slitherlink grid.

• (Repost because I misinterpreted the given Slitherlink grid) For those of you solving this, I've transcribed both grids to Penpa - let me know if there are any mistakes in the transcription
– HTM
Apr 11, 2021 at 3:11
• Also, I think a more compact way of saying the red square condition is that numbers in red cells (modulo 4) are liar clues for the Slitherlink, as in the number of edges around those cells are not correctly clued
– HTM
Apr 11, 2021 at 3:22
• Can you please change the colors you're using? Those particular shades of red and green are very bad for colorblind people - making one darker or lighter would make it much more readable.
– Deusovi
Apr 11, 2021 at 6:35
• I forgot about colorblindness when I was formatting the puzzle. I decided to change it to red and white, since green wasn't really important. Apr 11, 2021 at 10:37
• I would like to apologize. Apparently it was unsolvable and I made a mistake. I corrected the 9 in the middle of the first row to now be white. Apr 25, 2021 at 12:22

What an interesting and rewarding puzzle. There is indeed a mistake in the puzzle, but one that is easily rectified. It was a bit tedious doing this on paper, I wish there was an app to integrate something like this together. Anyway, the first step is obviously to start solving the sudoku. I will not post every single step, just the key ones that will allow everything else to fall together. After basic sudoku solving techniques I have the following:

Putting it into slither link we get this. Notice I omitted the incorrect numbers, I can work on those separately.

Here, there are 3 keys insights:

First of all cells I6 and I7 are both truth tellers. I6 has to be either a 0 or a 1, which means it can only be 1, 4, 5, 8 or 9. It cannot be 1, 4, 8 or 9 by sudoku rules so it has to be a 5 (1 in slitherlink).
After filling it in as a 1, I7 can only be a 2 or a 3, so it can only be 2, 3, 6, or 7. It cannot be 2, 3 or 6 so it has to be a 7 (3 in slitherlink).
Finally, cell D2 cannot be a 0. If it was a 0 then the 3 in C1 would not have an exit, it would self-loop which is not allowed. From sudoku column 2 #4 can only be in D2 or E2. D2 is a truth cell and cannot be a 0, so 4 cannot be in D2 so it must be in E2.

Once you plug in the new clues we just got back into sudoku you get this:

Looks pretty much solved already, here I went back and forth between the two puzzles a bit and there wasn't any more insightful hints. The final sudoku looks like this:

And the final slitherlink is:

Ok, notice the slitherlink isn't exactly completed. The problem is

Cell A5 is a red 9, which means in slitherlink it is NOT 1 line, which would imply going up and right. However if you do that then 2 in A7 will only have 1 line, and A7 is a truth teller. If you go right and up, A7 is satisfied but A5 is now telling the truth when it is in fact red. An easy solution is of course to either change A5 to white or A7 to red.

Let me know if I missed something!

• I swear that I looked over the puzzle several times. The intended solution is that A5 should be white. I will go edit the puzzle so it is more fair to people in the future. Apr 25, 2021 at 12:14