The resilient triangle

This triangle uses nine blue tiles. On the first row place A1=1, B1=2, C1=3 and D1=4. On the first column add A2=2, A3=3 and A4=4. Complete the diagonal with C2=4 and B3=4. All other cells can remain empty to study only this structure and follow its three lines clearly.

The row A1–D1 is 1–2–3–4 and so is the column A1–A4. The diagonal D1, C2, B3, A4 contains four 4s. The coordinates show why it is continuous: move one column left and one row down at each step. There is no curved line or jump between vertices to imagine.

The row and column meet at A1. The row and diagonal meet at D1. The column and diagonal meet at A4. Thus every pair of lines intersects, but at three different cells. None of those vertices belongs to all three lines at once.

This distinguishes pairwise intersection from global intersection. Merely checking that “the lines touch” would incorrectly suggest the shape can be defended. The correct criterion asks whether one particular tile belongs to every line. In the triangle none does: the opponent may break two lines by removing a vertex, but always leaves the third alive.

If Red removes A1, the row and column break, but the diagonal of 4s remains complete. Removing D1 breaks the row and diagonal, but preserves the column. Removing A4 breaks the column and diagonal, but preserves the row. Even removals that destroy two lines leave a winning one.

Removing an interior tile such as B1 destroys only the row; the column and diagonal survive. Removing C2 destroys only the diagonal. Any additional blue tile outside the shape would be still less useful for defense. This covers both vertices and interior cells and directly verifies the intersection criterion’s conclusion.

Count the values: one 1, two 2s, two 3s and four 4s. That totals nine tiles with no number exceeding its four copies. The 4s at D1 and A4 are shared by the diagonal and the straights, but they are individual tiles: belonging to multiple lines does not make them count twice in the reserve.

This check prevents drawing impossible tactical shapes. For example, two distinct matching lines of the same number would require more than four copies and cannot coexist in this game. In the triangle, combining straights with a matching group lets a tile serve multiple geometric roles without inventing extra material. Numeric legality matters as much as shape.

The triangle proves that two disjoint lines are sufficient but not necessary for resilience. It also teaches you to seek a third line avoiding the first two lines’ critical crossing. That question can generate useful candidates in games: “Which line could I complete without passing through the tile that currently breaks everything?”

Recognizing the finished shape does not guarantee you can build it from any position: the opponent has opportunities to remove tiles as it develops. Practice recognizing it under rotations and reflections, check values and reserve, and explain which line survives each critical removal. Then look for incomplete positions where your two placements can finish the shape before the opponent defends again.