Crowns Puzzle Guide: One Crown Per Region

Idea 1: a crown rules out everything around it

When a crown is placed, every other square in its row, its column and its region is out, and so are the eight squares touching it. Mark those squares × immediately. Tap a square once for ×, again for a crown, and a third time to clear it. Being generous with × marks is what makes the next two ideas visible.

The strongest single placements come from small regions. A region of one square must hold a crown: place it first and mark everything it eliminates. A region of two squares that are side by side eliminates the squares touching both of them.

Idea 2: confinement

If every remaining square of a region lies in one row, then that row's crown is in that region, so every other square in that row (outside the region) is out. The same holds for columns. This is the move that cracks most boards. Look at each region in turn and ask: are all its remaining squares in a single row or a single column? If yes, clear the rest of that line.

The reverse also works. If all the remaining squares of a row lie in one region, that region's crown is in that row, so the region's squares in other rows are out.

Idea 3: counting

Take any set of rows, say rows 1 and 2. They contain exactly two crowns. If exactly two regions fit entirely inside rows 1 and 2, those two regions supply both crowns, and any other region's squares inside rows 1 and 2 are out. The same works for columns, and for larger sets: three rows holding three complete regions exclude every other region from those rows.

Counting is the technique that feels advanced but is really just arithmetic. Scan the top rows and the left columns for regions that fit inside a few lines.

The touching rule, used positively

Because crowns cannot touch, two adjacent rows' crowns must be at least two columns apart. If a region's squares in one row are all next to a region's squares in the row below, the two crowns cannot both be in those squares... unless they are far enough apart. More usefully: a region shaped like a 2 × 2 block can hold only one crown anyway, and whichever square it uses, the squares touching all four are excluded. When two candidate squares in a row are adjacent, the square directly between them in the next row is out, because it touches both.

A method for a whole board

1. Place crowns in single-square regions and mark their eliminations.
2. For every region, check confinement to a row or a column.
3. For every row and column, check whether its remaining squares all lie in one region.
4. Count: find sets of rows or columns that exactly contain whole regions.
5. Whenever a row, column or region is down to one open square, place the crown.
6. Repeat. Each placement creates new confinements.

If you reach a point where every region has two or more open squares spread over two or more lines, go back to counting with sets of two or three lines; the board always has a next step.

Common mistakes

• Forgetting diagonal touching. The eight neighbours are out, not just four.
• Marking too little. Every deduction should end with × marks, which are the input to the next deduction.
• Placing a crown because a region has "mostly" one row. Confinement needs every remaining square in the line.
• Ignoring columns. The same reasoning in the other direction often breaks a stuck board.
• Guessing on large boards. A guess that fails has to be undone square by square. Counting is faster.

Glossary

• Region: a coloured area of the board that must hold exactly one crown.
• Confinement: when a region's open squares all lie in one row or column, so that line's crown is in the region.
• Open square: a square not yet marked × or crowned.
• Counting argument: matching n lines to n regions that fit inside them.