Star Battle

Rules

1.Place stars so that there is exactly one star in every row, exactly one in every column, and exactly one in every outlined region. There are always as many regions as rows.

2.No two stars may touch: not horizontally, not vertically, and not diagonally.

3.There is exactly one arrangement that satisfies all of this.

If you know Star Battle with two stars per row: here it is one, everything else is the same.

The five rules of deduction

Every puzzle in this book can be solved with the five rules below and nothing else. The difficulty label counts how many starting points lie open at the tightest moment: Easy four or more, Medium two or three, Hard exactly one, and you have to find it.

Mark ruled-out cells with a dot; it is the only way to keep track.

•A. Last cell. If a row, a column or a region has only one cell left that is not ruled out, the star goes there.

•B. What a star rules out. A star empties its whole row, its whole column, its whole region and the cells around it, up to eight of them. Do this the moment you place a star.

•C. A region inside one line. If all free cells of a region lie in one row, the star for that row must be inside the region, so rule out the rest of the row. The same holds for a column.

•D. A line inside one region. The reverse is also true: if all free cells of a row lie inside one region, that region's star is in this row, so rule out the region's cells elsewhere. Again the same for a column.

•E. Pigeonhole. If the free cells of k regions together occupy exactly k rows, those k rows belong to those regions alone: every other cell in them is out. Works for k = 2 and k = 3, with rows or columns, and mirrored: k rows whose free cells come from exactly k regions.

How to solve

1.Look for A first, and apply B the moment you place a star.

2.When that stalls, hunt for C and D. Look at small regions first: a region of two cells always lies inside one line.

3.When that stalls too, go to E: first k = 2, then k = 3.

Work down the list. A hard grid rarely opens with A: look for C, D or E first.

Worked example

A 6×6 grid, Easy. Solve along with the text. Every rule from the list above turns up here. The gray letters name the regions; they are only for this explanation and do not appear on the puzzle pages.

Nothing simple works yet. No row, column or region is down to a single free cell, so rule A finds nothing, and no region sits inside one line, so C finds nothing either. That is the normal opening of a Star Battle.

Whenever you place a star below, rule B immediately empties its row, its column, its region and the eight cells around it. That is not repeated every time.

Step 1, rule E. The free cells of regions E and F all lie in rows 5 and 6. Those two rows belong to those regions alone, so every other cell there is out. Out: r5c4, r5c5, r5c6, r6c5, r6c6.

Step 2, rule D. All free cells of column 5 lie inside region A, so that region's star is in this column, and its cells elsewhere are out. Out: r1c3, r1c4, r1c6, r2c4, r2c6, r3c6.

The grid now looks like this. A dot marks a cell that is ruled out, a star a placed star.

Step 3, rule A. Column 6 has only one free cell left, so the star goes on r4c6.

Step 4, rule A. Column 4 has only one free cell left, so the star goes on r6c4.

Step 5, rule C. All free cells of region C lie in row 3, so that line's star must be inside the region, so the rest of the line is out. Out: r3c3.

Steps 6 to 9, rule A throughout. Nothing clever is left. Each star you place empties enough cells that the next unit falls to a single free cell, and that one to the next: Column 3 to r2c3, Region A to r1c5, Region C to r3c1, Region E to r5c2.

The finished grid: