Sudoku

Rules

1.Fill the grid with the digits 1 to 9.

2.Every row contains each digit exactly once.

3.Every column contains each digit exactly once.

4.Every one of the nine heavily outlined 3×3 boxes contains each digit exactly once.

5.The printed digits stay as they are. The puzzle is solved when every cell is filled and rules 2 to 4 hold.

How to solve

The solver that gave every Sudoku in this book its difficulty label knows the 6 techniques below, and nothing else. It never guesses, and a puzzle it cannot finish was thrown away instead of being labelled. The label is relative to exactly this list: if you know more techniques, a Hard puzzle will feel easier than the label promises.

1.Naked Single, first needed at Easy. A cell with only one candidate left. Cross a digit off every cell that shares a row, a column or a box with it; where a cell has just one candidate left, write it in.

2.Hidden Single, first needed at Medium. A digit that fits into only one cell of a row, a column or a box. It belongs there even if that cell still carries other candidates.

3.Naked Pair, first needed at Hard. Two cells of one unit that carry exactly the same two candidates. Those two digits are used up by those two cells, so strike both from every other cell of that unit.

4.Hidden Pair, first needed at Hard. The mirror image: two digits that fit into only the same two cells of a unit. Then those two cells hold nothing else, so strike every other candidate from them.

5.Pointing, first needed at Hard. All remaining places for a digit inside one 3×3 box lie in a single row or column. The digit sits in that box, so strike it from the rest of that row or column outside the box.

6.X-Wing, first needed at Hard. A digit has exactly two possible cells in each of two rows, and those cells stand in the same two columns. Either way the digit takes opposite corners of that rectangle, so strike it from the rest of both columns. The same works with rows and columns swapped: two columns that share the same two rows.

Work technique 1 until it stalls, then 2, and so on. After every success go back to the top: a single new digit usually re-opens the cheap techniques.

Worked example

A Hard grid. The small digits are pencil marks, the candidates a cell still allows. Writing them in is the whole preparation; everything after that is crossing them out.

56 of the 59 steps are naked and hidden singles, and those need no explanation: a cell with one candidate left, or a digit that fits nowhere else in its unit. Shown below are the 3 places where that is not enough.

Step 16, Pointing. In box 9 the 7 appears only in row 8. Wherever it lands inside the box, it takes that row, so the 7 leaves the rest of row 8.

Step 17, X-Wing. The 2 sits in exactly two cells of column 3 and in exactly two cells of column 8, and both pairs share rows 1 and 3. Those four cells form a rectangle: whichever way it falls, the two rows are taken, so the 2 leaves them everywhere else.

Step 18, Naked Pair. r3c1 and r3c2 hold nothing but 3 and 4. Those two digits belong to those two cells, so they leave the rest of row 3.

The finished grid:

One more technique: the hidden pair

This one did not come up in the grid above, so here it is in another Hard puzzle, at its step 24.

In row 1 the 2 and 5 appear in no cells but r1c1 and r1c5. Those two cells must take those two digits, so every other candidate there is gone.