Centroid guess — find the center of mass of a shape

HOW TO PLAY

Slice the shape into two equal halves with one straight line.

Drag across — your stroke must fully cross the shape
Release to place the line; drag the endpoints to fine-tune
Press Confirm to score the cut

Math: by the Intermediate Value Theorem, every region has a 50/50 bisector in every direction — perfection is always reachable. Read more →

Cut in a random target ratio (5/95 … 50/50) shown up top each round.

Same controls as Half — one line, fully across
Aim to match the small-piece percentage

Math: for any r ∈ (0, 1) and any direction there's a line producing that exact ratio — IVT applied to the moving half-plane.

Two cuts that cross inside the shape — four equal pieces.

Draw two lines, each fully crossing the shape
Drag any endpoint to adjust, then Confirm

Math: Courant–Robbins theorem — any planar region can be split into 4 equal pieces by two perpendicular lines. Proof rotates two area-bisectors by 90° and uses IVT. Read more →

Two cuts, but the second must stay in one half — three equal pieces.

First line splits the shape; second splits only one of the halves
If the second line crosses the first inside the shape you get 4 pieces — invalid

Math: apply IVT twice — first find a 1/3–2/3 bisector, then halve the larger piece. Always achievable.

The cut's angle is fixed. Slide the line to find the 50/50 spot.

Drag anywhere — the line translates perpendicular to its direction
Endpoints snap to the outline automatically

Math: for any direction there is exactly one line of that direction bisecting the area — uniqueness and existence both from IVT.

Place four points on the outline to form a square — the closer to a perfect square, the better.

The nearest point on the outline follows your cursor
Tap / click to drop — drag points any time to adjust
After four points, press Confirm to score

Math: Toeplitz's Inscribed Square Problem (1911) — does every closed curve contain 4 points forming a square? Proven for polygons, smooth curves, and piecewise-smooth curves like the ones here; for arbitrary Jordan curves it's still open after 110+ years. Read more →

Find the largest equilateral triangle with all three vertices on the shape’s outline.

Same controls as Square — tap, drop, and drag to adjust
Score = regularity (equal sides, 60° angles) × size relative to the maximum inscribed equilateral
After three points, press Confirm to score

Math: every Jordan curve contains inscribed equilateral triangles — proven by Nielsen & Wright (1990) and others. Unlike Toeplitz's square, the triangle case is fully settled. Read more →

Find the center of mass of the shape — tap anywhere to place your guess.

Shapes with holes shift the center of mass
Tap anywhere on the board to drop your guess
Drag the point to fine-tune, then press Confirm

Math: the centroid doesn't have to lie inside the shape — for an annulus it sits at the empty center. Non-convex outlines and holes are where intuition breaks. Read more →

Slide a pole under the shape so it balances.

Tap anywhere to place the pole's X position
Drag to fine-tune, then press Confirm

Math: only the horizontal offset between the pole and the centroid's X matters for tipping — it's an inverted pendulum whose torque is m·g·Δx.

Place the shape on the pyramid tip so it balances.

Drag the shape to translate, use the purple handle to rotate
The shape must touch the tip, then press Confirm

Math: for any 2D shape there is always an orientation that balances on a single point — the Intermediate Value Theorem applied to the horizontal offset of the bottom point from the centroid as the shape rotates.

HOW TO PLAY

STATS

CHANGE PUZZLE