this post was submitted on 12 Feb 2025
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This looks like two lines intersecting at a 90° angle. So we have 3 variables we can control: the x and y coordinates of the intersection and the angle (modulo a quarter circle).
It feels like it’s always possible to distribute a population equally in the four squares. Anyone wanna prove it?
I can't find it now but I seem to remember a proof that a collection of points could be divided in half by a straight line, first by assuming that no three points were collinear, then by picking any point on the plane, drawing a straight line through it, then rotating it around that point until you could get half the points on each side. The implication of this though is that you could pick someone in Aberdeen, which the above map would seem to suggest isn't possible.
The location of the second line would have to be determined by sliding it along the first, rather than rotation, so it could end up resting on 0, 1 or 2 points. Either way you're probably close enough. The proof, for or against, is outside my mathematical ability.
https://en.wikipedia.org/wiki/Ham_sandwich_theorem