Why seven are not enough

A consecutive straight of length n contains n−3 four-cell windows when n is between 4 and 8. Four tiles give one window; five give two; six give three. Each additional endpoint tile adds a window, but that window may still depend on cells supporting every earlier one.

To see this, write 1–2–3–4–5–6–7 on A1–G1. The windows start at A1, B1, C1 and D1. All contain D1, the fourth tile in the sequence, with value 4. There are four completed lines, yet one removal at D1 destroys all four. Visible length can conceal a very simple defense.

In a straight of four, any of its four cells breaks the single window. With five, the middle three break both. With six, the middle two break all three. With seven, only the middle cell breaks every window. Check each case by sliding a window of four and noting which cells appear in every window.

The pattern helps both defense and attack. When defending, avoid instinctively removing an endpoint: interior windows may survive. When attacking, a seven-tile straight can force the opponent onto one particular cell, but it remains vulnerable. That obligation may have tempo value; it is not victory against a correct defense.

The sequence 1–2–3–4–5–6–7–8 on A1–H1 contains five windows. Their global intersection is empty. A particularly short proof uses just two: A1–D1 and E1–H1 are valid disjoint straights. Every removal leaves one intact. The intermediate windows add lines but are unnecessary to prove resilience.

The eight-tile straight uses one copy of each number and comfortably respects the inventory. It also fits in a full column or an eight-cell main diagonal. A legal winning shape once finished is not necessarily something you can build without resistance: every opposing turn includes a removal that may interrupt your progress.

Eight is the minimum for any resilient shape, not just a row: no position with fewer than eight own tiles preserves a line against every removal. If two disjoint lines exist, they already occupy eight cells. If the lines cross, you must eliminate any shared point; with too few tiles there is insufficient geometry to achieve that.

The subtle case involves lines that intersect pairwise. If they all lie on one straight geometric line, their four-cell intervals share a cell. If two lie on different geometric lines, they share one point and contain seven tiles together. A third avoiding that point while meeting both must add at least two new tiles: that makes nine. This rules out seven or fewer.

After completing your own kth placement turn, you have k+1 tiles on the board. You therefore cannot reach eight before your seventh own turn: global turn 13 for Blue or 14 for Red. The resilient win would be confirmed during the next opposing removal. This is a possibility bound, not a promise you can force that position.

A win caused by a defensive error can occur earlier: the opponent might preserve a single line by removing elsewhere. The eight-tile minimum concerns surviving every legal defense. Keeping these cases separate prevents treating a short game as a refutation of the proof or confusing a bad defense with an unavoidable strategy.