A curved chase with four participants and a symmetry that is not broken at any time.
The four turtles
Riddle statement
Four turtles occupy the four corners of a square of 10 centimeters on each side.
They all start to move at the same time and at the same speed.
Each one always advances in the direction of the turtle in front of it, continually turning to aim at it.
How far does each turtle travel before the four turtles meet?
Show solution
Solution
By symmetry, the four turtles always continue to form the vertices of a square that shrinks as it rotates.
Let's look at a turtle A and the next one, B.
A always moves directly towards B. The distance between them decreases exactly at the speed with which A advances in that direction, because B moves perpendicular to that instantaneous line of pursuit.
That means that the separation between A and B is reduced at the same speed with which A travels the path.
As the initial distance between consecutive turtles is 10 cm, A will have traveled exactly 10 cm when that distance is cancelled.