Cage elimination
If a digit appears in none of a cage's possible combinations, it can be removed from every cell of that cage.
What it is
A cage usually allows several digit sets, not just one. Collect all sets that still reach the sum given the digits already placed, then take their union. Any digit outside that union cannot appear anywhere in the cage and can be struck from all its cells at once.
How to spot it
Look at cages of three or four cells whose sum is well away from the middle of the range. Then check which digits appear in none of the remaining combinations.
Worked example
Step through the example. First the position, then the cage the argument rests on, then the candidates, and finally what follows from it.
Step 1 of 4: The starting position
- Pattern cells
- 1/4, 2/4, 3/4
- Digits
- 1
- Follows from it
- 6/4, 7/4, 8/4, 9/4
Press a marked square to point it out on the board above.
Why it matters
Only the listed combinations can produce the sum. A digit not present in any of them therefore cannot occur in the cage, regardless of arrangement. That makes it a genuine elimination, not a guess.
Practise now
The practice puzzle is picked so that this technique is what gets the solve moving again.
Frequently asked questions
How do I list the combinations without a table?
Start from the smallest possible set and work up. For a three-cell cage of 8 that is 1+2+5 and 1+3+4 — so 6, 7, 8 and 9 are impossible.
Does it still apply when some cells are filled?
Yes, and it gets stronger. Each placed digit removes combinations, so the union of the remaining ones shrinks.
What is the difference from cage combinations?
Cage combinations tell you which sets are possible. Cage elimination draws the conclusion: digits in no set can be struck.
Put it to use
Where you will meet it
Hard grids are built around moves like this one. That is where it earns its keep.
Play HardStuck on a grid of your own?
Type the position into the solver and it walks the whole solution, naming the technique behind every step — including this one.
Open the solverFinished reading? Mark it done and the learning path remembers where you got to.