The New Sudoku Players' Forum

[Spiffy McBang]

I'm having an issue with Cracking the Cryptic's Classic Sudoku #23 (5/10 star difficulty). This is where it stands:



(If the image doesn't load, here's a link: https://imgur.com/m5WFzTT)

I was stuck enough to check the hints. Here's what the last one says:

"You can use the Empty Rectangle technique on box7 (and also in other places in the grid!). In box 7, the number 7 must either appear in r8 or (if the 7 was in r9c1) in c1. But note that, if the 7 is in c1 then, because 7 must go in one of two positions in r2 (either c1 or c5), it MUST then be in c5. So either way, r8c5 cannot contain a 7 and must be a 9."

My current board state would seem to align with the first part of this hint. The options for 7 in box7 are in r8 (c1 and c2) and r9c1. And, of course, if the 7 is in c1, then the 7 in r2 would have to be in c5, like in an x-wing.

But as it stands, nothing that I can see is stopping the 7 from being in r8c2. If that were the case, then r9c1 would be a 9, r8c1 would be a 5, and r2c1 would be a 7, leaving r8c5 to also contain a 7. I went so far as to put it in a sudoku solver that explains what it's doing to solve the puzzle, but the only clue it gave was that the 5s in row 8 meant there couldn't be a 5 in r9c1, which I already have noted. It solved the puzzle otherwise through brute force, which I'm trying not to do.

I must be missing something, but I can't figure out what. Am I misunderstanding how the empty rectangle works? Have I messed up something else which renders the empty rectangle non-functional (e.g. there should be more restrictions in r8c2 that would put everything into place)? For a while I stared at r4, since it has a slightly more complicated setup than all the other rows, columns, and boxes, but I couldn't find a logical way to work the puzzle through there, either.

[thelardoffear]

Current state of the puzzle.

Code: Select all

+-----------+---------+--------+
|6   1   79 |79  2  5 |8  3  4 |
|579 4   3  |8   79 1 |2  6  59|
|8   59  2  |6   3  4 |7  1  59|
+-----------+---------+--------+
|2   3   579|479 6  79|1  45 8 |
|1   79  8  |479 5  3 |49 2  6 |
|4   6   59 |1   8  2 |59 7  3 |
+-----------+---------+--------+
|3   2   1  |5   4  8 |6  9  7 |
|579 579 4  |2   79 6 |3  8  1 |
|79  8   6  |3   1  79|45 45 2 |
+-----------+---------+--------+

If you look at row 2, 7 must be in R2C1 or R2C5.
If R2C5 = 7 then R8C5 <> 5
If R2C1 = 7 then in box 7, R8C2 = 7 which means R8C5 <> 7.

In general Empty Rectangles work as follows:

Code: Select all

...|... If A & B are the ONLY positions for a digit in a row and ? & X are
.A.|.B. are only positions in a box (not all need actually be available).
...|...
---+--- If A is digit then * isn't digit.
...|.?. If B is digit then either of Xs must be digit so * isn't digit.
.*.|X?X So we can eliminate *.
...|.?.

[storm_norm22]

sometimes its good to see the image with some lines drawn so you can see the connections to the candidates.
the 7's in column 5 are strongly inferenced as well as the 7's in box 7. you can see this strong inference in the image with the solid line in column 5 and the solid line in box 7. and the weak links with the hollow lines.
written in Eureka we have (7)r2c5 = (7)r8c5) - (7)r8c12 = (7)r89c1
what this chain proves is that the 7 in r2c5 and the 7's in r89c1 are strongly inferenced. therefore, the poor 7 in r2c1 is now a victim of logic, and can be eliminated.