OK, I'll explain how this particular XY chain works in words and refer you to some teaching sites that have good general explanations and nicely worked examples.
An XY chain consists of a string of bi-value cells, where one candidate is common to adjacent cells in the chain and the adjacent cells can see each other.
The idea is to start and end on the same candidate, and you can then eliminate that candidate from cells that can see the first and last cells in the chain.
So, with this one, start by assuming that cell a is not 1, then it must be 8. So cell b can't be 8, so it must be 2. So cell c can't be 2 so it must be 9. So cell d can't be 9, so it must be 1.
So, summarizing this, if cell a is not 1, cell d is 1. You then reverse the argument and start with cell d and assume that it is not 1 and following the cells in order d-c-b-a you will conclude that cell a is 1.
What all this proves is that at least one of cell a and cell d must be 1. Since the target elimination cell, which I've marked with a * in the diagram can see both cells a and d it can't be 1.
This particular XY chain is of length 4 but they can be of just about any length. The shortest useful XY chain is of length 3 and is called an XY Wing, which you may have heard about.
The teaching site XY chain examples can be found
here and
here and just for good measure you can also read about XY Wings on the same sites
here and
here.
Leren
PS An alternative solution is to first apply the X Wing, as Jason suggested, as shown in the following diagram : The X Wing is in 1's in rows 68 / c34 in the cells marked * and eliminates the 1's in r2c3 and r5c4 as shown.
- Code: Select all
*-----------------------------------------------------*
| 7 4 5 | 2 8 9 | 6 1 3 |
| 16 19 69-1 | 5 3 7 | 4 2 8 |
| 2 3 8 | 4 6 1 | 9 7 5 |
|-----------------+-----------------+-----------------|
| 3 18 2 | 6 4 58 | 15 9 7 |
| 168 7 4 | 39-1 19 58 | 15 36 2 |
| 9 5 *16 |*13 7 2 | 8 36 4 |
|-----------------+-----------------+-----------------|
| 5 29 7 | 8 29 6 | 3 4 1 |
| 4 28 *19 |*19 25 3 | 7 58 6 |
| 18 6 3 | 7 15 4 | 2 58 9 |
*-----------------------------------------------------*
This brings you to here, where a length 3 XY chain ie an XY Wing in cells a, b and c, solves r9c1 and the whole puzzle. I suspect that this was what the magazine had in mind as the preferred solution.
- Code: Select all
*-----------------------------------------------------*
| 7 4 5 | 2 8 9 | 6 1 3 |
|a16 19 b69 | 5 3 7 | 4 2 8 |
| 2 3 8 | 4 6 1 | 9 7 5 |
|-----------------+-----------------+-----------------|
| 3 18 2 | 6 4 58 | 15 9 7 |
| 168 7 4 | 39 19 58 | 15 36 2 |
| 9 5 16 | 13 7 2 | 8 36 4 |
|-----------------+-----------------+-----------------|
| 5 29 7 | 8 29 6 | 3 4 1 |
| 4 28 c19 | 19 25 3 | 7 58 6 |
|*8-1 6 3 | 7 15 4 | 2 58 9 |
*-----------------------------------------------------*
An alternative move at this point, suggested by Jason, is to apply a BUG+1 move, which forces r5c1 to be 1. If you really wan't know why this is True, read about the BUG principle
here .
Leren
