Theme

Decided by exhaustion — page 13

Questions with finitely many cases, settled by going through all of them — and what changes when a claim about every argument becomes a count.

Ten points, thirteen halving lines. points 0.253,1.518 0.242,0.216 0.717,0.119 -0.427,0.222 0.554,0.409 0.299,0.346 0.806,0.006 -1.101,0.118 1.209,-0.414 0.445,0.714; halving pairs 0-9 1-2 1-3 1-5 2-4 2-6 3-5 3-7 4-5 4-9 5-6 5-9 6-8; degrees 1,3,3,3,3,5,3,1,1,3.

Topology

Lines that leave half on each side

Scatter ten points on a page and draw every line through two of them that leaves four on each side. There are at least five such halving lines and at most thirteen, and every point lies on an odd number of them. Both facts come from turning a line through half a revolution, the argument that halves two shapes with one cut. How many halving lines n points can have is the k-set problem, and its answer lies somewhere between a little more than n and n to the four-thirds.

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