Star battle: rules, techniques and boards to play

Tap a cell once to put a dot in it, which means "no star here", and tap again for a star. A third tap clears it. On a keyboard, move with the arrow keys and press Enter. There are 4 boards at 8×8 and 4 at 10×10.

The rules

• Every row has exactly two stars.
• Every column has exactly two stars.
• Every colored region has exactly two stars.
• No two stars touch, side to side or corner to corner.

That's it. If you've played the Queens puzzle, it's the same game with one crown per line. Two stars sounds like a small change, but it means you can't just look for the one cell left in a row. You have to think about pairs.

How these boards are made

I made these with my own generator. It lays out a legal set of stars, grows a region around each pair, then reshapes the regions until only one answer is left. A separate solver then checks every board again: it has to find exactly one answer, and the five techniques below have to reach that answer with no guessing. A board that needs trial and error doesn't make it onto the page.

Five techniques

Each example below is a real moment from one of the boards above, drawn just before the step. The outlined cells are the line or region the step is about, a green star is a new star, and a cross is a cell that just became empty.

1. Full lines

Once a row, column or region has its two stars, every other cell in it is empty. And every cell touching a star is empty too. This sounds obvious, but crossing these cells off as you go is what makes the other techniques visible.

2. Only one way to fit them

Count how many stars a region still needs, then look at where they could go. Two stars can't touch, so a small region often has only one way to hold both. A region of three cells in a row, for example, has to use both ends.

3. Cells no star can use

Sometimes a region has a few ways to place its stars, and one of its cells isn't in any of them. That cell is empty, even though you don't know yet where the stars go.

4. Cells every option touches

This is the same idea, pointed outward. If every way of placing a region's stars puts a star next to some cell outside it, that cell can never hold a star. Look for it below and beside narrow regions.

5. Regions boxed into rows

If all the open cells of a region sit in one row, the region's two stars are that row's two stars, so the rest of the row is empty. With two regions squeezed into the same two rows, they take all four of those rows' stars, and every other cell in those rows is empty. It works the same with columns, and the other way round: if two rows can only get their stars from two regions, those regions have nothing left for anywhere else.

A few habits that help

• Start with the smallest regions. They have the fewest ways to hold two stars.
• Any 2×2 square holds at most one star, because all four cells touch each other. Two rows next to each other on an 8×8 board split into four of those squares and need four stars, so each square gets exactly one.
• Dot every cell you rule out. Most mistakes come from forgetting a cell that was already empty.
• When you're stuck, go back to technique 5 and count regions against rows and columns.

Questions

Queens
One crown in every row, column and region. 680 boards with one answer each and hints that explain every step.

Get it on Google Play