This math puzzle is because of Donald Knuth (so far as I do know; perhaps he were given it from somebody else) circa 2014.
Believe a undeniable represented via the unit sq.. In this undeniable we need to “peacefully encamp” two armies of point-sized squaddies — one military purple and one military inexperienced. Each and every soldier “assaults” chess-queen-wise: horizontally, vertically, and diagonally in all instructions. The puzzle is to maximise the dimensions of the equivalent armies (equivalently, maximize the dimensions of the smallest military), given the constraint that no pair of opposing squaddies will also be positioned attacking each and every different.
(No Cantor units or anything else, k? Simply standard polygons.)
I’ve a conjectured resolution however I do not know if it is truly the optimal. If this could be extra on-topic for math.SE or one thing, please remark and let me know!
I’ve written a bit Javascript program to assist visualize answers. You’ll in finding it right here.
Listed here are two examples of how to peacefully encamp armies of sub-maximum dimension. The whole dimension of each and every military within the first answer is 1/9; the overall dimension of each and every military in the second one answer is 1/8.
If you’ve solved that, the following puzzle is to peacefully encamp 3 armies, 4 armies, and so forth… the entire technique to infinity.
My very own candidate answers have military dimension $frac{7}{48} approx 0.1458$ (for two armies), $sim 0.0718$ (for three armies), $0.05$ (for 4 armies), and $sim 0.0311$ (for five armies).
