Split a shape into four equal pieces
Draw two straight cuts that each fully cross the shape and intersect inside it. Together they carve the shape into four pieces, and you’re scored on how close those pieces are to one quarter of the total area each. Drag either line’s endpoints to adjust, then confirm.
The math: Courant & Robbins’ four-piece theorem
A classical result in What Is Mathematics? by Courant and Robbins proves that any planar region can be divided into four equal pieces by two perpendicular straight lines. The proof is beautiful: for every angle θ, there’s an area-bisector in that direction (from the Intermediate Value Theorem). Fix one bisector, then rotate a second bisector perpendicular to it by 90° — by continuity, somewhere during the rotation the two bisectors cut the region into four pieces that are all equal.
The puzzle doesn’t force perpendicularity — any two crossing bisectors work as long as each one splits the shape in half on its own and each also splits the other’s two halves in half. That’s a tighter constraint than it looks, and it’s what makes this variation feel so different from the simple half cut.
Tips for four equal quarters
- Place your first line as a rough 50/50 bisector (same intuition as Half Cut).
- Then place a second line perpendicular-ish — perpendicular isn’t required, but it’s often the easiest guess.
- Slide the second line until the two pieces on each side of the first line look equal.
- If a piece looks much bigger than the others, the crossing point is off-center toward that piece — shift both lines away from it.
Related
- Half Cut — one line, two equal halves.
- Tri Cut — two cuts, three pieces.
- Target Ratio — hit a specific non-half ratio.
- Constrained Angle — cut at a locked angle.