Diagonal Sudoku

Because the diagonals add constraints, the stored puzzles start with fewer givens than classic ones, an average of about 19 against about 24.5, and the board offers four levels, Easy to Expert.

What exactly is added?

Draw a line from the top-left corner to the bottom-right corner, and another from the top-right to the bottom-left. Each line passes through nine cells, and each set of nine must hold the digits 1 to 9 once. Nothing else changes: rows, columns and 3×3 boxes work as in classic Sudoku. The rules page restates this with the other variants, and the controls page covers input and settings.

The diagonals reach every box. Each of the four corner boxes holds three cells of one diagonal, the center box holds three cells of each, and the four edge boxes hold none. In the game, cells on a diagonal are highlighted along with their row, column and box when you select them, so you can see the extra constraint at work.

How many cells does each cell see?

On a classic board every cell has twenty peers. Here the number depends on where the cell sits:

• A cell that is on neither diagonal still has twenty peers.
• A cell on one diagonal, such as a corner, has twenty-six.
• The center cell, which is on both diagonals, has thirty-two.

That is why a digit placed on a diagonal is so informative: it clears itself from many more cells than a digit placed elsewhere. When you start a puzzle, look at the cells on the diagonals first, since they carry the most constraints.

What do the stored puzzles look like?

The site has 150 puzzles at each of Easy, Medium, Hard and Expert, and no Evil puzzles, since every generated puzzle could be solved at Expert or below. Clue counts run from 16 to 23, with averages between 19.1 and 19.7 across the levels. A classic puzzle averages 24.4 to 24.8. That is roughly five fewer givens, which fits with the extra constraint doing some of the work the givens do on a classic board. Each puzzle is verified unique before every release.

Which techniques appear?

From the technique-frequency data for this board, with 150 puzzles per level:

• Easy: singles only.
• Medium: pointing pairs in 99% of puzzles, box/line reduction in 45%, naked pairs in 40% and hidden pairs in 20%.
• Hard: naked triples in 68% and X-Wing in 39%, on top of the Medium techniques.
• Expert: simple coloring in 91% of puzzles and XY-Wing in 44%, with Swordfish in 1%.

The Expert figure for simple coloring is the standout. On the classic board it appears in 57% of Expert puzzles. Here it is the dominant technique, so if you want to prepare for Expert diagonal puzzles, learn simple coloring first. The reason is not established by the data. A plausible explanation is that the diagonals create more two-cell links for a digit, which is what coloring uses, but this page does not test that.

How should you solve it?

1. Treat the diagonals as two more houses. When you scan for a hidden single, include them. A digit that fits in only one cell of a diagonal is a hidden single.
2. Check the center cell last. It sees thirty-two cells, so it usually has few candidates by the time you reach it.
3. Use the diagonals in pointing-style arguments. A digit confined within a box to cells that lie on a diagonal can be removed from the rest of that diagonal.
4. Keep your notes current. Placing a digit on a diagonal removes it from up to thirty-two cells, so automatic note removal is worth leaving on. See the pencil marks guide.

What mistakes are common here?

The usual one is forgetting a diagonal when counting what a cell can hold. A cell that looks as though it has three candidates may have two once you check the diagonal it sits on. The common mistakes guide lists this one with the other variant-specific slips.

Where next?

If the diagonals feel comfortable, try hyper Sudoku, which adds four extra regions, or anti-knight, which restricts cells instead of adding houses. If you want arithmetic instead of lines, killer Sudoku is the contrast. The games hub lists every board, and the strategy guide, strategy by difficulty, shows what to learn at each level.