Fair Division · Interactive

Eight Fair Slices

Every polygon can be sliced fairly — but only in certain counts. Slide the sides and pieces to find the forbidden numbers, watch one measurement rebuild the whole, then step through the puzzle’s own solution.

The Fair-Slice Machine

Choose a regular polygon and a number of pieces; the machine cuts it into equal-area triangles and measures every piece live. But watch the tick strip: some counts are teal, and some are simply… not allowed. The forbidden ones aren’t hard — they’re impossible, by theorem.

Sides of the polygon — 3

Number of equal pieces — 8

Impossible —

Reshuffle the cuts

One Measurement Rebuilds the Whole

Every fair slice knows the size of the entire polygon — that’s the engine of the puzzle. Set the length of the marked stretch (the rust piece’s base above) and watch the reconstruction, step by step, with the trigonometry cancelling as it goes.

The marked stretch measures — 7

The Puzzle, Step by Step

The original eight-slice triangle, solved slowly. Each step lights up the part of the figure doing the work.

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Piece areas are verified live with the shoelace formula. The forbidden counts are real theorems: Monsky (1970) bars every odd count for the square; Kasimatis (1989) restricts every regular polygon of five or more sides to multiples of its side count. Triangles alone accept every number — and in more ways than one, as the reshuffle shows.

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