Hashi Strategy: Capacity First, and Never Seal a Suburb

🧮Capacity arithmetic — the whole opening in one idea

An island's neighbours are the islands it can see up, down, left and right with no island in between. Each neighbour can take at most 2 bridges, so an island with n neighbours has capacity 2n. Compare its number to its capacity:

• Number = capacity: every direction is a double bridge. Rare and wonderful.
• Number = capacity − 1: every neighbour gets at least one bridge (if any direction were empty, the rest couldn't reach the number). Place the singles, then see what's left.
• One neighbour only: everything goes there — a 1 is a single, a 2 is a double, done.
• A corner 4 (two neighbours, capacity 4) is doubles both ways; a side 6 (three neighbours) and a middle 8 are the same capacity-equal case — doubles everywhere the moment you spot them.

🧪Worked example — three openings on one real board

From the Wednesday, October 14, 2026 DailyGrid board (10×10):

The loneliest 2. The 2 at the top (row 1, column 6) can see exactly one island — the 4 directly below it in row 6, with nothing in between. One neighbour, number 2: double bridge down. First move of the day, no thought required.

The facing 1s. Down the left edge sit two 1s, at rows 6 and 8 of column 1, in clear view of each other. They can never be connected to each other: two 1s joined together form a finished two-island network, and the rules demand one network for the whole board — they'd be sealed off forever. So each 1's bridge goes the only other way it can: right, one bridge each (to the 4s in column 3). On this board both really do.

The squeezed 3. The 3 at row 6, column 8 has only two neighbours (the 4s to its left and right) — capacity 4, number 3, so by the capacity − 1 rule both neighbours get a bridge, and one of them gets the double. Which one isn't decided yet — and that's fine. Draw both singles now and let the 4s' own arithmetic settle the double later.

🌐The connectivity rule — Hashi's soul

Every completed island group must stay capable of joining the whole. Concretely: never make a bridge that seals a finished group off from unfinished islands. Before confirming any bridge that would complete the last island of a cluster, trace the cluster's other connections — if it has none and islands remain elsewhere, that bridge (or that double) is wrong; one of its bridges belongs somewhere else. This single check resolves most "stuck" positions: the move you're missing is usually the one that doesn't close a suburb.

✖️Crossings and leftovers

• A drawn bridge blocks the crossing line — an island whose only remaining neighbour lies across a bridge is a contradiction; work backwards from it.
• Count what's left, not what's placed. An island needing 1 more with two open directions is a coin the neighbours will flip for you; an island needing 2 with one open pair left is a forced double.
• Finish islands in pen. When an island hits its number, mentally (or literally) close its remaining directions so you stop offering it bridges it can't take.

❓FAQ

What are the forced moves?

Number equals capacity → doubles everywhere; number equals capacity − 1 → at least one bridge each way; one neighbour → all bridges there.

Can two neighbouring 1s connect?

Only if that pair is the whole puzzle. Otherwise they'd seal themselves off — each must connect elsewhere.

Where are the basic rules?

In the companion guide: How to Play Hashi.

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