The intersection decides

Two lines share a tile when they include the same physical cell. Using the same number is not enough, nor is having their extended geometric lines cross outside their four cells. A horizontal A4–D4 and a vertical D1–D4 do share D4. The reserve and values must allow that tile to serve both.

If those are your only lines, D4 is a complete defense: removing it eliminates both. Two orientations do not automatically create two independent threats. This is a crucial difference between recognizing shapes and understanding resilience. Count which tiles support the lines as well as how many lines you can draw.

The global intersection is the set of cells present in every completed line you have. With one line, it contains all four cells. With two crossing lines, it may contain only the crossing. With two disjoint lines, it contains none. With three or more lines, keep comparing: the first two lines’ intersection may disappear when the third is included.

If at least one line exists and the global intersection is empty, every removal leaves some line alive. If the intersection contains cells, the opponent can remove any of them and break every line. Always start by checking that a completed line exists: several unfinished projects do not yet force the opponent to make a particular defensive removal.

Removing a tile destroys exactly the lines containing it. Lines that exclude it keep their four cells, their values and their color. To destroy every line with one removal, the selected tile must belong to all of them. This proves both that a shared cell permits defense and that without one no removal can defend.

The proof does not depend on matching groups versus straights, orientation or distance. It only requires lines that are already valid on the final board. It also does not assume a reasonable opposing choice: it covers every legal removal. That makes it an exact immediate-victory rule rather than an approximate evaluation score.

Consider 1–2–3–4–5–6 on A1–F1. Its windows are A1–D1, B1–E1 and C1–F1. The first two share B1, C1 and D1; including the third leaves only C1 and D1. Either removal breaks all three windows, even though the row shows many tiles and several combinations.

This procedure prevents the mistake of choosing the tile present in the most lines without checking whether it belongs to all of them. A tile in three of four lines may look most influential, but removing it loses if the fourth remains. In a mandatory defense, the threshold is every line, not a majority.

When comparing placement pairs, ask how the intersection changes. Adding a line that also passes through the only critical point may increase the line count without improving resilience. Adding a line that excludes that point can eliminate the common defense. The difference lies in shared cells rather than the visual size of the shape.

Do not conclude that disjoint lines are the only way to win. They are sufficient and easy to prove. With three lines, every pair can touch at a different place while no cell belongs to all three. The triangle chapter constructs exactly that case with legal values and reserves.