Why are 2s not possible in these Sudoku cells?

Question

I recently was playing Sudoku on my iPad. As I was stuck on the puzzle I clicked on "Tip" to receive a hint on how I can solve this Sudoku (I am still learning the game) - so please don't judge me for this.

It added all kinds of pencil marks - and I was unable to follow why some of them were not populated. Unfortunately I do not have a before & after picture. I have marked the 2's in different colors - of my current understanding of the game. Can somebody explain, why the red ones were not marked?

Answer 1

This tool looks pretty sophisticated! I think each red circle might have slightly different reasons for not containing a 2 hint, but as an example, let's look at the red circle on Row 1, Column 2.

Suppose we did write a 2 there. Then:

• To complete Column 2, we must write a 4 in Row 3 (it can't go in Row 8 because of the box) and a 3 in Row 8.
• From there, we know Row 8, Column 5 must be 6 (all other numbers are already aligned with the cell in at least one direction).
• Which solves Row 7: Column 4 contains a 3 and Column 7 contains a 6.
• This in turn solves Row 5: Column 7 contains a 3 and Column 9 contains a 6.
• And it finishes off Column 9 because we know to write a 3 on Row 3.
• But now we are in trouble: where, on Row 1, can we write a 3?

It can't go in Column 2 because we started by writing a 2 there. It can't go in Column 4 because we added a 3 to Column 4 on Row 7. It can't go in Column 6 or Column 7 either, and that's all the open columns. Since we arrived at this point by following the only possible logical path beginning with a 2 at Row 1, Column 2, we've "proven" that writing a 2 there would be a mistake.

I haven't tried the other cells you circled in red in the image, but I imagine following a similar algorithm leads each choice into a contradiction or error of some kind.

Edit: Importantly, each of the other choices along this path are valid on their own (which is why the numbers indicated in these steps also tend to show up in those hints); for example, we might be able to put a 3 on Row 8, Column 2, without making it impossible to solve Row 1, as long as the cell on Row 1, Column 2 remains open. It's the combination of all these steps, beginning with writing a 2 on Row 1, Column 2, which leads to the unsolvable scenario at the end.

Answer 2

My theory: the 2s were removed arbitrarily in order to help you (since you are left with cells with only one candidate). Obviously, if you search for a reason for that removal, you will eventually find one just by solving the puzzle.

Why do I think they were removed arbitrarily?

1. Jesse's (excellent) answer shows that the reason for that removal is very convoluted. I was not able to find a simpler reason.
2. There are candidates that were not removed by the "tip" button. This shows that the logic behind the candidate removal is quite limited, which contradicts 1.
3. There are easier moves for you (moves that allow to fill cells) before even removing those 2s

What are the candidates that were not removed?

In cell (7,1) 3 and 6 should be removed because both are either in (7,5) or (7,7)

What are the easier moves?

First row: you cannot place a 4 in (2,1). That would mean a 2 in (7,1), a 6 in (6,1) and a 3 in (4,1). Both 2 and 4 should then go into (6,2).

You have a 3+6 chain that goes from (9,3) to (5,8). That means that if you put a 3 in (9,3) you get a 6 in (5,8), and conversely. If 3 and 6 go in (9,3) and (5,8), neither of them can go in the intersection cells, which are (9,8) where you have already put a 1, and (5,3) where there should then be a 1.

With that latter move, finishing the puzzle is quite straightforward.