Logic Puzzle Club

The break-in: Queens of the week 4

Queens of the week 4 is a 10×10 that solves in 15 steps. Steps 1 to 7 are routine. Step 8 is the first deduction that isn’t routine, which solvers call the break-in.

By James Murray · Updated

Last square in a region

Two regions are a single square each, so their queens go straight in. A third region has three squares, and the queen in row 2 rules out two of them, so its queen goes in row 3, column 4.

Step 1. Last square in a regionThe region at row 5, column 1 has one square left that can hold its queen, at row 5, column 1. Its queen goes there.
Step 2. Last square in a regionThe region at row 2, column 10 has one square left that can hold its queen, at row 2, column 10. Its queen goes there.
Step 3. Last square in a regionThe region at row 2, column 4 has one square left that can hold its queen, at row 3, column 4. Its queen goes there.

Regions that own a row

When every open square of a region sits in one row, that row’s queen has to be in the region. Everything else in the row is out.

Step 4. Region in one rowEvery open square of the region at row 1, column 1 is in row 1. So row 1’s queen belongs to that region, and the rest of row 1 can’t have one.
Step 5. Region in one rowEvery open square of the region at row 1, column 5 is in row 4. So row 4’s queen belongs to that region, and the rest of row 4 can’t have one.

Two suppositions

Each step picks a square and supposes a queen there. That queen would rule out every open square of some region, so the square can’t hold a queen.

Step 6. Ruling out by touchingSuppose a queen sat at row 9, column 3. It would rule out every open square of the region at row 6, column 1, which then has nowhere for its queen. So row 9, column 3 can’t have a queen.
Step 7. Ruling out by touchingSuppose a queen sat at row 10, column 6. It would rule out every open square of the region at row 8, column 6, which then has nowhere for its queen. So row 10, column 6 can’t have a queen.

The break-in

Two regions, the top-left one and the one along the left edge in rows 6 to 10, now have every open square in columns 2 and 3. Two columns hold exactly two queens, and these two regions need both. So no other square in columns 2 and 3 can have one.

Step 8. Two regions in two columnsThe open squares of the regions at row 1, column 1 and row 6, column 1 all sit inside columns 2 and 3. Those two columns hold exactly two queens, and the regions need all of them. So no other square in columns 2 and 3 can have a queen.

After the break-in

Steps 9 to 15 each find a region down to its last square. The first two are drawn here.

Step 9. Last square in a regionThe region at row 2, column 1 has one square left that can hold its queen, at row 10, column 5. Its queen goes there.
Step 10. Last square in a regionThe region at row 6, column 1 has one square left that can hold its queen, at row 9, column 2. Its queen goes there.

The rest of the solve

The whole solve is 15 steps, each one forced. See every step in the walkthrough, or play the puzzle yourself first.

Play week 4The full walkthroughThis week’s Queens