_{Previous variant}

Today’s Sudoku variant is a bit weird. Here’s why:

Take this screenshot:

From this, you might argue that there is nothing that can be logically deduced on this grid. Now you might be right concerning normal Sudoku rules, but, here’s the thing: Numbers on a line must either be one less or one greater than a number right next to it. With this, we can deduce that there must be a 2 at R3C2, a 3 at R3C3, and a 4 at R3C4, as you can see with this next screenshot:

Then I guess it could be logically deduced that there is a 5 that goes in R3C1 (due to Sudoku), although that’s as much that could be logically deduced before hitting a dead end with what could be logically deduced.

So, the gimmick basically is that numbers on a line must either be one less or one greater than a number that is next to it on the same line. But what does it mean for a number to be on the same line?

Take this next example:

You might not be able to tell at first that this line, with no surrounding digits, has 2 different ways that it could be filled in. However, here’s a question that you might have: Couldn’t we just have a 3 at R4C4? It’s part of the same line and is right next to the 2, which is also on the same line.

The answer is no. Ignoring the fact that this leads to an impossible grid state, the 3 would only be connected due to the fact that the line splits in the middle of the 4 cells, and grid connections based off of that just aren’t allowed. In fact, here is a screenshot with the actual legal grid placements based off of this example:

Note that R3C3 could either be a 1 or a 3 since either number would be a valid number without any numbers to help logically deduce what it is.

Here is how the grid is split up in a 6×6 Sudoku for reference:

Box 1: R1C1, R1C2, R1C3, R2C1, R2C2, R2C3
Box 2: R1C4, R1C5, R1C6, R2C4, R2C5, R2C6
Box 3: R3C1, R3C2, R3C3, R4C1, R4C2, R4C3
Box 4: R3C4, R3C5, R3C6, R4C4, R4C5, R4C6
Box 5: R5C1, R5C2, R5C3, R6C1, R6C2, R6C3
Box 6: R5C4, R5C5, R5C6, R6C4, R6C5, R6C6

The puzzle:

Play online!

To get the $\color{green}✓$:

1. Solve the Sudoku grid:
2. Show the steps you took to solve it.