Sudoku Coach

When a cell's row, column and box already hold eight different digits, the ninth is its only candidate and goes in that cell.

A digit with only one possible cell in a house goes in that cell, whatever other candidates the cell has. To find one, pick a box and a digit it is missing, then follow that digit through the rows and columns that cross the box.

If two cells in a house have the same two candidates and nothing else, those cells will take both digits, so neither digit can go anywhere else in that house. Three cells that together hold only three candidates form a naked triple and work the same way, even when no single cell holds all three.

When two digits can only go in the same two cells of a house, those cells must hold those two digits, so every other candidate in them can be removed. A hidden triple is three digits confined to the same three cells, and the same rule applies.

If every candidate for a digit inside a box lies in one row or column, the box's copy of that digit will land in that line. Nothing else in the line can take it, so remove the digit from the line's cells outside the box.

This is pointing turned around. If every candidate for a digit in a row or column lies inside one box, the line's copy of the digit is in that box, so the box's other cells cannot take it.

Take one digit. If two rows each have exactly two places for it, and those places line up in the same two columns, the digit fills opposite corners of that rectangle. Both columns then get their copy from those two rows, so remove the digit from the rest of both columns. Swap rows and columns and it works the same way.

A swordfish extends the X-wing to three rows. If a digit's candidates in three rows all fall within the same three columns, with two or three in each row, those rows supply the digit for all three columns. Remove it from those columns in every other row. Swap rows and columns and it works the same way.

An XY-wing uses three cells with two candidates each. The pivot holds X or Y and sees two pincers, one holding X or Z and the other Y or Z. Whichever digit the pivot takes, one of the pincers becomes Z, so any cell that sees both pincers cannot be Z.