Sudoku’s rules are four sentences

A sudoku looks like a number puzzle, but there is no arithmetic in it. The digits are just labels. You could use letters or colors and nothing would change.

What makes it a puzzle is four sentences. Every cell holds exactly one digit. Every row holds each digit exactly once. So does every column, and so does every box. Each sentence is a “for every” with an “exactly one” inside it.

Every technique in the next level is one of these sentences put to work. When a solver says “this cell can’t be 5”, they are quoting a rule.

“Exactly one” packs two promises together. At least one: row 3 has a 1 somewhere. At most one: row 3 never has two 1s.

The two halves do different jobs. “At most one” rules digits out: row 3 already has a 4, so no other cell in row 3 can be 4. “At least one” puts digits in: if only one cell in row 3 can still be 1, the 1 goes there.

Row 3 below shows both. Its 4 rules 4 out of the other three cells. And its 1 has only one place left, because columns 1 and 3 already have a 1. So row 3, column 4 is 1.

Without sentence four, a cell could hold two digits, or none. The row rules alone can’t stop that. Solvers lean on it all the time: when a cell has only one digit left that it could be, sentence four says it must be that one.