PQRS and QRTU are two faces of a cube with PQ = 12.
A beam of light emanates from vertex P and reflects off face QRTU at point X, which is 7 units from QU and 5 units from QR. The beam continues to be reflected off the faces of the cube.
The length of the light path from the time it leaves point P until it next reaches a vertex of the cube is given by A√B, where A and B are integers and B is not divisible by the square of any
prime.
Find A and B.
The components of the beam's travel when it reaches X are 12, 5 and 7, making that first leg, before reflection have length sqrt(218).
As in a hall of mirrors, we can consider the problem as being one straight line in an infinite array of tessellated cubes.
Each passage through a plane parallel to QRTU has a motion with a 12 component, a 5 component and a 7 component. But also the multiple of 5 and the multiple of 7 must also be a multiple of 12 so that all axes have experienced a multiple of 12 units since leaving P. The numbers 5, 7 and 12 are relatively prime, so the number of segments of motion will be 5*7*12 = 420, each having length sqrt(218).
So the answer is 420 * sqrt(218) ~= 6201.22568529803.
The only fly in the ointment is that the beam will pass through edges (in the actual world, hit a 2plane corner several times). Ideally it would bounce straight back visavis the two planes without affecting the results. Real world mirrors might just absorb light where two mirrors join.

Posted by Charlie
on 20160301 15:22:28 