From the people behind the Alan Turing Cryptography Competition.
You are reading the website of the 2019 edition of the MathsBombe Competition, which ended on Sunday 28th April at 11:59 pm

Puzzle 6

The long corridor which leads to the Piccabonus Bank's main vault consists of two flat vertical walls parallel to each other, the North Wall and the South Wall, which have a mirror finish. A security laser shines its beam from the point $$N_0$$ on the North Wall, gets reflected off the South Wall and is detected back at $$N_0$$.

An earthquake happens and shifts the South Wall which is still flat and vertical but is no longer parallel to the North Wall. As a result, the laser beam shining from $$N_0$$ is now reflected off the South Wall at $$S_1$$, then off the North Wall at $$N_1$$, then off the South Wall at $$S_2$$ and so on (see diagram).

Trying to assess the damage, Engineer Ada uses an interferometer and finds that, since the laser beam leaves $$N_0$$, it travels exactly 332.8935m until it hits the North Wall for the 50th time at $$N_{50}$$. Then it travels exactly 662.687m between $$N_{50}$$ and $$N_{100}$$.

Ada needs to know the length, in metres, of the path the laser beam travels between $$N_{100}$$ and $$N_{150}$$. But due to health and safety concerns, it is impossible to move the equipment to $$N_{150}$$. Calculate the theoretical value of the length of that path.

Note: the answer is a finite decimal, and you have to enter all the significant figures.