# Sudoku variants #2: Lined Numbers (6x6 grid)

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 6x6 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.
• Fun fact: this puzzle is solvable even if all the lines going between different boxes are removed and any one digit is given instead of the given four. Commented Nov 28, 2023 at 5:56

Solution:

Step by step:

1:

To start, if there is a 1 on a line, there must be a 2 next to it. If the number along the line from the 2 is then in the same box, line or column as the original 1, then the next number must also be a 3, and similarly for other numbers. Using this, we can immediately fill out 3 boxes, as they are all along a line.

2:

Now we have filled out all the boxes that we can, look at the first column. The 4 cannot connect to another 3, so it must be a 2, and that lets us complete the column. In fact, the 5 and 4 placement now lets us fill in the bottom left box as it is all along a line.

3:

The 1 in the bottom left box now must connect to a 2 along the line, and to the right of that 2 must also be a 1. Following the line up from the newly placed 2, we get a sequence of forced entries due to one of the two candidates being ruled out from being in the same column or row, and we can complete the entire line. This leaves just one lone cell to be filled, which simply must be a 6.