Theme

Counting the same thing twice — page 7

One collection, counted by two different methods, and an identity that falls out because both answers have to agree. The proof is the pair of counts.

One experiment, measured by runs and by awakenings. Two unit squares for the same coin and the same schedule: the same experiment weighed two ways: by runs, heads keeps half the square; by awakenings, heads is one of 3 equal slices.

Probability

One coin, counted by runs and by wakings

Beauty is put to sleep and a fair coin is tossed. Heads, she is woken once; tails, twice, with the first waking erased from her memory. Each time she wakes she is asked how likely heads is. One half, say some; one third, say others; and unlike every earlier puzzle of this kind, stating the protocol exactly does not end the argument.

Every way to pair the edges of a hexagon. Chord diagrams of all 15 pairings of a 6-gon's edges, shaded by the surface each gluing makes: 5 spheres, 10 tori.

Topology

Every way to pair a polygon's edges

A hexagon's six edges can be paired in fifteen ways. Glue each pair head to tail and five of the fifteen give a sphere and ten give a torus; an octagon's 105 pairings give 14 spheres, 70 tori and 21 surfaces with two handles. The spheres are exactly the pairings whose chords never cross, and the whole table obeys one recurrence found in 1986.

A random pairing of 24 edges. Chord diagram of one uniformly random pairing of a 24-gon's edges, corners coloured by the vertex they become. a 24-gon with its edges paired at random and glued head to tail: the 24 corners fall into 3 vertices, so the surface has genus 5, against a most possible of 6.

Topology

The surface a random gluing makes

Pair the edges of a large polygon at random and glue each pair head to tail. The surface almost always has nearly as many handles as the polygon allows: a thousand edges leave about seven and a half vertices, and the genus is within four of its ceiling of 250. The vertices behave like the cycles of a random permutation, and their average is a harmonic number.

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