Double-cat strategy · step 4 of 4

Band counting

Two rows hold exactly four cats. If two whole colors live inside those rows, the four cats are spoken for — and every square of any other color in the band is a cross. This is the signature move of 2★ Star Battle.

Blue and gold fit entirely inside the top two rows. Two colors × two cats = four cats, and two rows hold exactly four — so green’s lone toehold in the band dies immediately.

The argument is pure arithmetic, and it runs in one breath: the top two rows will end up with four cats; blue and gold must place all four of theirs somewhere in there; four claims on four slots leaves zero for anyone else. No cat has to be placed, no square examined — the shapes of the colors alone hand you the cross.

It works in every size: three rows hold six cats, so three colors trapped in three rows evict everyone else; a single row holding a whole color means that row’s two cats are that color’s two cats. And it transposes to columns — scan bands of columns for colors that don’t escape them.

The count runs both ways

The mirror argument is just as useful: if the open squares of a band of rows all belong to a small set of colors whose remaining cats exactly match the band’s remaining slots, then those colors have no cats to spare elsewhere — cross out their squares outside the band. When the numbers match, the band and the colors lock each other in place.

Placed cats just adjust the arithmetic. A band of two rows that already holds one cat has three slots left; a color that already placed one cat only claims one more. Keep the running totals and the technique survives deep into the endgame.

How to actually spot it

• Scan the board’s small, flat colors — wide-but-short shapes that fit inside one or two rows (or tall-thin ones inside a column or two). They are the usual band-makers.
• Any time two such colors overlap the same two rows, do the four-count. It takes five seconds and pays off absurdly often on 10×10 double-cat boards.
• This is the double-cat sibling of one-cat confinement: same idea of shapes trapped in lines, upgraded from “one region, one line” to honest-to-goodness counting.

MeowSolver reports these as Locked set hints — at double-cat the message spells the count out (“must place all 4 of their remaining cats in rows 3 & 4”), and the locking colors are outlined so you can verify the trap.

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