Sudoku techniques · Extreme
ALS-XZ in Sudoku
ALS-XZ links two Almost Locked Sets, groups of cells with one more candidate than cells, through a restricted common candidate X that can only be placed in one of them. That forces a second shared candidate Z into the other, so Z can be removed from any outside cell that sees every Z in both sets.
Why it works
Two almost-locked sets linked by a restricted common candidate force another shared candidate into at least one of them, so that candidate can be removed from any cell outside the two sets that sees every cell with that candidate across both sets.
How to spot it
The hard part is seeing the sets, not the logic.
- An ALS is any group of cells in one unit with one extra candidate: a single bivalue cell is the smallest one.
- Two sets, two shared candidates: X must be restricted, meaning every X in one set sees every X in the other. Z must not be.
- Eliminate Z from cells outside both sets that see all the Zs in both.
Builds on
Worked example
These are the app's own lesson boards. Step through them the same way a hint does.
- 1
Look at the first set . These 2 cells all lie in the same row, column, or box and have 3 different candidates among them, one more than the number of cells. This is called an Almost Locked Set (ALS).
- 2
Now look at the second set . These 3 cells all lie in the same row, column, or box and have 4 different candidates among them, one more than the number of cells, so they also form an ALS. The two sets share candidate 3 as a restricted common candidate (RCC): every cell with candidate 3 in one set sees every such cell in the other, so the digit 3 can be placed in at most one of the two sets, never in both.
- 3
Both sets contain 5 as a candidate. Since the digit 3 can be placed in at most one set, at least one set will not use 3. A set that does not use 3 has as many distinct candidates left as cells, so it must place each of them, including 5, somewhere inside it. So 5 is placed in at least one of the two sets, and we can remove the candidate 5 from any cell outside the two sets that sees every cell with candidate 5 across both sets. Here X is 3 and Z is 5.