geometric.games

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.

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.

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 and smooth curves; for arbitrary Jordan curves it's still open after 110+ years.

Place three points on the outline to form an equilateral triangle.

Same controls as Square — tap, drop, and drag to adjust
All three sides must be equal and every angle 60°
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.

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.

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.

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