Three-piece triangle-to-square dissection allowing flips

Establish whether there exists a dissection using exactly three polygonal pieces that transforms an equilateral triangle into a square when reflections (flips) of pieces are allowed in addition to translations and rotations, without overlap; or prove that no such three-piece dissection exists under these rules.

Background

The paper proves that no dissection with three or fewer polygonal pieces exists between a square and an equilateral triangle when flips (reflections) are forbidden. This resolves the century-old question of whether Dudeney’s famous four-piece dissection is optimal under the no-flip model.

However, the role of flipping remains unresolved. Allowing reflections changes the matching constraints between piece boundaries and can potentially enable configurations that are impossible without flips. Determining whether the three-piece impossibility persists in the flip-allowed setting would close a key gap in the dissection landscape.