Intersections and elimination
Every white cell sits in two runs. It needs a digit both of them allow. When that isn’t enough, test each set and strike the ones that can’t fit.
How it works
Each white cell is in one across run and one down run. Its digit has to be in a set that makes the across clue and in a set that makes the down clue. That shared list is the intersection.
When one run is a unique combination and the other is short, the shared list is often a single digit. That is the fastest way into most grids.
Harder grids need one more step: elimination over whole sets. A set is out when the run’s cells can’t hold its digits. Strike it, and the digits only it used go too.
Why intersection works
The cell’s digit is in the across set, and it is in the down set. So it is in both: that is just “and”. Anything on only one list is out.
How to eliminate a set
Write the run’s sets. For each one, ask whether its digits can go into the cells one each, using only digits the cells still allow. If a digit of the set has nowhere to go, the whole set is out.
This is proof by contradiction on a small scale: suppose the run used 4 + 6 + 8. The bottom cell would need 6 or 8, but it is 5 or less. That breaks a rule, so the set is out.
Worth memorizing
A cell’s digit fits both of its runs.
Take the digits both sets allow.
A set is out when its digits can’t all be placed.
Check each digit has a cell that allows it.
Start from a unique run and follow its crossings.
Each crossing is a new intersection.
Practice
0 of 4 rightThe ringed cell is in across 23 and down 16. What goes there?
What goes in the ringed cell?
The ringed cell is in across 8 and down 16. What goes there?
The top cell is 4 or 5 and the bottom cell is 5 or less. Which set is left for down 18?