Slice any shape into two equal halves

Draw a single straight line that fully crosses an irregular shape and splits it into two pieces of equal area. A fresh shape is generated every round from a seeded random outline — sometimes smooth and convex, sometimes jagged with inward notches and holes. Your cut is scored by how far the smaller piece is from exactly 50%. Drag the endpoints after placing the line to fine-tune, then confirm.

The math: why a perfect half always exists

By the Intermediate Value Theorem, every bounded region has a 50/50 bisector in every direction. Sweep a line of a fixed angle from one side of the shape to the other: one extreme has 0% of the area to its left, the other has 100%. The area-to-the-left function is continuous, so it must pass through exactly 50% somewhere in between. Change the angle and you get a different bisector — so every shape has infinitely many perfect half-cuts, and the puzzle is always solvable.

The trickier part is doing it by eye. Human intuition gets fooled by long skinny limbs, by holes that shift the visual balance, and by shapes that are wider than they are tall. Learning to ignore the outline and focus on where the area sits is the whole game.

Tips for getting perfect cuts

  • Find the longest axis of the shape and cut roughly perpendicular to it — limbs along that axis dominate the area.
  • Watch for holes: an empty region effectively subtracts area from one side.
  • Use the endpoint drag: start with a rough cut, then slide the endpoints until the two pieces look equal by eye.
  • If a piece has a long thin tail, it’s lighter than it looks — a compact lobe outweighs a long one of similar length.

Other cut variations

Once the 50/50 cut feels natural, try the variations. Each tests a different flavour of geometric intuition:

  • Target Ratio — hit an exact non-half ratio like 37/63.
  • Quad Cut — two crossing cuts, four equal quarters (Courant–Robbins).
  • Tri Cut — two cuts, three equal pieces.
  • Constrained Angle — the cut’s angle is fixed, slide it to the sweet spot.

Or switch mode entirely: inscribe a square (the unsolved Toeplitz problem) or balance the shape on a pole.