Recognize every line

A line consists of exactly four adjacent cells in one fixed direction. To check A1–D1, read A1, B1, C1 and D1 without skipping anything. For a diagonal such as B2–E5, read B2, C3, D4 and E5. The other slope counts too: E2, D3, C4 and B5 are four consecutive diagonal cells.

You cannot turn halfway, form a corner or jump over an opposing tile. A visually connected shape is not always a line. Follow the entire direction with your eyes and separate recognizing the geometry from recognizing the numbers; this helps you spot diagonals that are easy to overlook.

There are two valid families. In a matching group, all four numbers are equal. In a straight, the difference between each number and the next is always +1 or always −1. Thus 3–4–5–6 and 6–5–4–3 are valid, whereas 3–5–4–6 and 3–4–3–4 are not. Four different numbers alone are insufficient.

The numeric endpoints are real boundaries. The number after 8 is not 1 and the number before 1 is not 8. Therefore 6–7–8–1 is not a straight. When completing three tiles, verify the required value at the exact position: a 5 can complete 2–3–4–5, but placing a 1 to the right of 2–3–4 does not work.

An incomplete line may have an interior gap. If A3 contains 2, B3 is empty, C3 contains 4 and D3 contains 5, B3 needs a blue 3 if the other tiles are blue. Completing a gap requires both the correct cell and the correct value; placing any tile next to the group is insufficient.

You can also fill two gaps in one turn. With a 2 at A3 and a 5 at D3, placing B3=3 and C3=4 completes the straight, provided B3 and C3 are empty and both tiles are available. Recognizing this prepares you for two-placement exercises without confusing a finished line with a resilient threat.

Across A1–E1, the sequence 1–2–3–4–5 contains two valid lines: A1–D1 and B1–E1. Each consecutive block of four is checked separately. The five tiles are not a new category of combination; they contain two ordinary combinations sharing B1, C1 and D1. Removing any of those three cells breaks both.

An 8×8 board has 130 four-cell windows: 40 horizontal, 40 vertical and 25 along each diagonal slope. You need not examine them as a list every turn. Start with those passing near the most recently placed tile, and remember that diagonals and overlapping windows deserve as much attention as obvious rows.

For each candidate, ask whether all four cells are occupied, whether they share one color and whether the values fit either family. Then list other windows that the same placement may have created. One tile can complete a horizontal and a diagonal line together, although that alone does not prove the position survives removal.

Practice describing a combination using coordinates and values: “B2=7, C3=6, D4=5 and E5=4.” That sentence lets you review geometry, orientation and reserve without relying on a visual impression. If an exercise answer surprises you, reconstruct that description; it usually reveals a jump, an ordering error or a window you had not counted.