Imagine four bugs situated at each vertex of a unit square.
Suddenly, each bug begins to chase its counterclockwise neighbour.
If the bugs travel at 1 unit per minute, how long will it take for the four bugs to crash into one another?
First, the author explains that the bug is travelling towards its neighbor and maintaining a 45 degree angle, with respect to the center of the square.
Second, the author explains that why are trying to find out the radial component of the bug to the center of the square.
Third, the author explains the radial speed was r, then r is the same as the length radial line of a vertex of the starting square, which has length 1 to the center. The actual value of r is square root of 2/2.
Although this puzzle is popularly solved using equations, this is the first time I am seeing this solved using graphical approach.
I do not understand why did he chose to use the diagonal line as a reference to the center of the square to calculate the radial component.
By symmetry, the four bugs always occupy the vertices of a square. The square shrinks and rotates as the bugs chase one another, but symmetry implies that its center does not move. Therefore the diagonals of the square always meet at the center of the unit square and the bug's radial velocity is the projection of the bug's velocity vector along the diagonal that passes through the bug's position.
The velocity vector of a bug always points to the next bug, and is therefore along a side of the square. The angle between that side and the diagonal is $45\unicode{xb0}$. The radial velocity is therefore $\frac{1}{\sqrt{2}}$ units per minute; the initial distance of the bug from the center of the unit square is also $\frac{1}{\sqrt{2}}$ units. Hence the four bugs meet at the center after $1$ minute.