The squares of two consecutive whole numbers add up to 365. What is the larger of the two squares?

Half of 365 is 182.5, and $13^2=169$ sits just below it while $14^2=196$ sits just above. Checking, 169 + 196 = 365, so the larger square is 196. (Algebraically $n^2+(n+1{)}^2=365$ gives $2n^2+2n-364=0$, i.e. $n^2+n-182=0$ and $n=13$.) The trap is 169 — that is the smaller square, and $225=15^2$ overshoots.