How do you solve hard Queens puzzles without guessing?

Hard boards need two more ideas. The first is counting: if two colours can only use the same two rows, they will take both rows' queens, so everything else in those rows goes (and the same for three colours in three rows). The second is the no-room test: if a queen on some tile would leave another row, column or colour with nowhere to place its own queen, that tile is impossible.

1. Pairs: two colours in two lines

You already know that one colour trapped in one line claims it. Pairs extend that. Two colours whose open tiles all sit inside the same two rows (or columns) must use one row each. Those two rows are now spoken for.

queenalready ruled outplay herecross out nowtiles to look at

The pearl colour and the rose colour only have open tiles in column 3 and column 4. Two colours in two lines: between them they will use every queen those lines have, so every other tile in column 3 and column 4 can be crossed out.

2. Triples

The same count works with three colours in three lines. Triples are rarer and harder to see, so look for them only when pairs have run out.

The emerald colour, the topaz colour and the lagoon colour only have open tiles in column 2, column 3 and column 1. Three colours in three lines: between them they will use every queen those lines have, so every other tile in column 2, column 3 and column 1 can be crossed out. Both are covered in Pairs and triples.

3. The no-room test

Pick a tile and imagine a queen on it. It rules out its row, column, colour and neighbours. If that would wipe out every open tile of some other row, column or colour, the queen can't go there.

A queen on the crossed tile would rule out every open tile in row 2, leaving it with nowhere to put its own queen. So that tile can't hold a queen. This is deduction, not guessing: you only look one step ahead, and the conclusion is certain. Full page: The no-room test.

Where to look first on a hard board

• The edges and corners. Colours squeezed along an edge are the ones that share two rows or two columns.
• Small colours next to each other. Two small colours stacked in the same two columns are a pair waiting to be noticed.
• Tiles that see a lot. A tile in the middle of a small colour's neighbourhood is the best candidate for the no-room test.

Getting faster

• Do the cheap checks first, every time: forced queens, then single colours trapped in a line, and only then pairs and the no-room test.
• After every queen, re-scan the lines and colours it touched. That's where the next forced move appears.
• Mark tiles as you rule them out. Holding eliminations in your head is where mistakes come from.

Want a daily test?

The Gauntlet is ten boards in a row, up to 10×10, and it's hardest on Sundays.