# Solving Sudoku purely by logic: why does this X-cycle choose this cell?

I've been reading several articles on difficult Sudoku puzzles and whether "guessing" is ever acceptable. The attached image shows the stage I reached using only deduction but at this point I could not see a purely logical correct next step. I eventually found the solution by "brute force" choosing one number from a pair and proceeding from there. (My first choice was the wrong one of course).

After posting this question I was introduced to a Sudoku solver, a fascinating insight into strategies I have never considered before. I could follow everything although whether I could manage it with just pen and paper is another matter. The only problem I have is at the step which requires use of the X-CYCLE. I cannot see how you decide that C9 is where we should try 4 as the potential solution. Once chosen I can see how the strategy works but why choose 4 in that Cell over 4 in any other cell?

• @JLee I used that solver just now (with the original numbers, not the handwritten numbers) and it came back with "Very Hard Grade Overall Score: 251". Perhaps you entered a number wrong? Jul 26, 2022 at 15:08
• According to this solver "Tough Grade Overall Score: 137 Comment: This puzzle could require strategies beyond its grade Note: a partially completed puzzle will have an easier grade than the original puzzle. Grading is most accurate if the number of clues is less than 30.
– JLee
Jul 26, 2022 at 15:09
• @IanMacDonald yep i did. Corrected now. I entered the numbers found so far, so that explains the difference between mine and yours. Does this solve path help?
– JLee
Jul 26, 2022 at 15:15
• Thank you for all your comments. The Solver is a fascinating insight into strategies I have never considered before. I could follow everythung although whether I could manage it with just pen and paper is another matter. The only problem I have is at the step which requires use of the X-CYCLE. I cannot see how you decide that C9 is where we should try 4 as the potential solution. Once chosen I can see how the strategy works but why choose 4 in that Cell over 4 in any other cell? Jul 27, 2022 at 9:11
• @PaulFisher edit that comment/question into your question. Maybe someone well-versed in sudoku strategies will answer.
– JLee
Jul 27, 2022 at 16:26

For me, this one is easier to see if I start with simple alternation elsewhere.

Let's consider all the squares where 4 is still a possibility:

I can see that there are a number of rows and columns where I have 2 choices for where to put the 4, meaning I have a strong correlation there. For example if Row 3 Column 4 is NOT 4, then Row 9 Column for MUST be, and vice versa.

I'm going to choose red and yellow to color my squares. These do not (yet) correlate to making a choice about whether or not a particular square holds a 4, it just says that whatever is correct for that square, its partner must have the opposite.

These are the only strongly linked pairs definite in this chain, but I can see that there is another chain possible, I'm going to color those orange and purple

Now I can start thinking about which squares are at the intersection of these chains, in particular, I notice these guys (the other intersections are either already solved, or not candidates for 4)

The square at the purple/ yellow intersection has too many inputs to be an easy check for if/then statements, but the red/orange intersection contributes a single input to each chain, so that's a good place to start checking.

Since it is at the intersection of 2 chains with strong connections on candidates of 4, I should first check to see if it can be 4. When I propagate that through the chains I have identified, I see that I get a contradiction.

• Many thanks for the explanation. It does make sense so I'll try to remember in future! Jul 28, 2022 at 7:47