Solution to Challenge 1

Method
1. Fold the square in half one way and unfold, then in half the other way and unfold. The two creases cross at the centre in four 90 degree angles.
2. Try 2 mountains and 2 valleys (for example, the horizontal line as a mountain and the vertical line as a valley). The centre will not close flat.
3. Try 3 mountains and 1 valley (flip one half-line the other way). The square now collapses flat into a folded stack.

Why
- Maekawa's theorem says a flat-foldable vertex needs |M - V| = 2.
- 2 mountains and 2 valleys give |2 - 2| = 0, so the vertex cannot fold flat.
- 3 mountains and 1 valley give |3 - 1| = 2, so it can. The same holds for 1 mountain and 3 valleys.
- The angles (90, 90, 90, 90) already pass Kawasaki's theorem (90 + 90 = 180 each way), so the creases' types are what decides this one.

Checkpoint: 3 and 1 folds flat, 2 and 2 does not.