R bloggers

Le Monde puzzle [#1037]

FavoriteLoadingAdd to favorites

A purely geometric Le Monde mathematical puzzle this (or two independent ones, rather): Find whether or not there are inscribed and circumscribed circles to a convex polygon with 2018 sides of lengths ranging 1,2,…,2018. In the first (or rather second) case, the circle of radius R that is tangential to the polygon and going through all nodes (assuming such a circle exists) is such that a side L and its corresponding inner angle θ satisfy L²=R²2(1-cos(θ)) leading to the following R code R=3.2e5 step=1e2 anglz=sum(acos(1-(1:2018)^2/(2R^2))) while (abs(anglz-2pi)>1e-4){ R=R-step+2step(anglz>2pi)(R>step) anglz=sum(acos(1-(1:2018)^2/(2R^2))) step=step/1.01} and the result is > R=324221 > sum(acos(1-(1:2018)^2/(2R^2)))-2*pi [1] 9.754153e-05 (which is very close to the solution of Le Monde when replacing sin(α) by α!). The authors of the quoted paper do not seem to consider the existence an issue. In the second case, there is a theorem that states…
Original Post: Le Monde puzzle [#1037]