MathDB

Let

ABCD

and

BCFG

be two faces of a cube with

AB=12.

A beam of light emanates from vertex

A

and reflects off face

BCFG

at point

P,

which is 7 units from

\overline{BG}

and 5 units from

\overline{BC}.

The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point

A

until it next reaches a vertex of the cube is given by

m\sqrt{n},

where

m

and

n

are integers and

n

is not divisible by the square of any prime. Find

m+n.

Problem Statement