Once you have a handle on how to solve Star Battle, the next leap in skill comes from learning to count. Not counting stars you have already placed, but reasoning about how many stars a group of cells can possibly hold given the no-touch rule. This technique — sometimes called capacity counting or the pigeonhole argument — unlocks deductions that pure elimination cannot reach on its own.

Why Counting Works

Every row, column, and region must contain exactly two stars, and no two stars may touch — not even diagonally. That second constraint is the engine behind counting. Because touching is forbidden, stars must stay separated by at least one empty cell in every direction. That spacing requirement caps the number of stars any contiguous run of cells can hold, and when that cap equals the number of stars still needed, you have found a forced placement.

If any of this feels new, it is worth revisiting the rules before going further — the no-touch constraint is subtle and easy to underestimate.

How Many Stars Can N Cells Hold?

Start with the simplest case: a straight line of cells in a single row. Because no two stars can be adjacent, you need at least one gap between every pair of stars. In a run of two cells you can fit at most one star. In a run of three cells you can fit two stars, one at each end — the alternating pattern star, gap, star, which carries on as the run grows. In general, a run of n cells in a line can hold at most ⌈n ÷ 2⌉ stars.

The diagonal constraint tightens this further when you move into two-dimensional areas. A 2×2 block of cells can hold at most one star, because any star placed there attacks all three remaining cells. A 3×3 block can hold at most four stars (the four corners), though row and column quotas usually cap it lower. The key habit is to ask: given this shape of cells, what is the maximum number of non-touching stars it could ever contain?

The Pigeonhole Argument: Regions, Rows, and Columns

The most powerful application of counting is the pigeonhole argument. Suppose you identify a set of regions whose cells, taken together, lie entirely within a set of rows of the same size. For example, three regions whose combined cells touch only three rows. Those three regions together need six stars (two each). But six stars spread across three rows also means exactly two stars per row — the full row quota — and every one of those six stars must sit inside those three rows. That means no other region can place any of its stars in those three rows.

This argument works in any direction. If two regions fit entirely within two columns, those four stars fill both column quotas, and every other region is pushed outside those columns. Spotting these containment relationships is the heart of counting at the intermediate and advanced levels, and it is one of the key ideas covered in the broader strategies guide.

A Worked Example in Words

Imagine a seven-by-seven puzzle. Regions A, B, and C each have cells scattered across the puzzle, but every single cell belonging to any of those three regions falls in rows one, two, or three. Regions A, B, and C together must place six stars. Rows one, two, and three together must also contain exactly six stars. Those two facts together mean all six of the row quota for rows one through three must come from regions A, B, and C — and, equally, all six stars that A, B, and C need must land in those three rows.

The immediate payoff: every cell in rows one through three that belongs to any other region can be eliminated. Those cells cannot hold stars, because the row quotas are fully spoken for by A, B, and C. You have just cleared a potentially large set of cells without placing a single star yet.

Counting Remaining Capacity After Placements

Counting also helps once stars are on the board. After placing stars, update your mental tally of how many stars each row, column, and region still needs, and how many valid cells remain available to receive them. If a region still needs one star and only one cell in that region is not yet eliminated, that cell must be a star — this is just elimination. But counting adds a layer: if a region needs two stars and the remaining valid cells are so close together that they would touch, you know something elsewhere in the puzzle must be wrong, and you can backtrack sooner rather than later.

More usefully, if a region needs one more star and only two valid cells remain, you can sometimes deduce which row or column that final star must fall in even before knowing the exact cell. That row or column constraint then ripples outward to eliminate cells in other regions.

Combining Counting with Elimination

Counting and elimination are complementary. Elimination removes cells one by one based on adjacency and quota completion. Counting removes entire swathes of cells by proving a group of regions monopolises a band of rows or columns. In practice, you will cycle between the two: count to clear a band, eliminate within the resulting smaller spaces, count again on the updated puzzle, and so on.

A useful routine is to look for near-containment first — regions that almost fit within a band of rows or columns, with only a small number of cells spilling outside. If eliminating those spill cells is already justified by other logic, the containment becomes exact and the pigeonhole argument fires. Train yourself to scan for these near-containments whenever you feel stuck.

Common Mistakes to Avoid

  • Forgetting diagonals. The no-touch rule includes diagonal adjacency. A counting argument that ignores diagonal neighbours will overestimate capacity and lead to wrong conclusions.
  • Applying containment to the wrong dimension. Check carefully whether your set of regions is contained in rows, in columns, or in neither. Containment in both dimensions simultaneously is rarer but even more powerful when it occurs.
  • Counting cells instead of valid cells. Once some cells are eliminated, only the remaining un-eliminated cells count toward capacity. Recalculate after every deduction.
  • Assuming a near-containment is exact. Double-check that every cell of every included region truly falls within the band before drawing conclusions. A single overlooked cell in an outside row breaks the argument.

Building the Habit

Like most puzzle techniques, counting becomes intuitive with repetition. At first, consciously run through the checklist: pick a region, find all its cells, note which rows and columns they touch, and ask whether a small set of regions is fully contained in an equally small band. Over time this scan becomes a reflex — you will start spotting containment relationships almost automatically while your eyes move across the grid.

Medium puzzles are an ideal proving ground for counting. They are complex enough that pure elimination stalls early but structured enough that one or two pigeonhole arguments will unlock the rest of the puzzle. Work through several at this level with counting explicitly in mind before moving up.

Put It into Practice

The best way to cement any technique is to use it on a real puzzle right now. Head to the puzzle page and pick one that has been giving you trouble. Before you make your next move, spend a minute scanning for region-band containments. You may be surprised how quickly the counting argument cuts through what felt like a dead end.