# From 2-D to 3-D

**Geometry**Level 5

In a cubical room in 3-D coordinate space a ball is projected from point (\(\frac { 1 }{ 4 } ,0,\frac { 1 }{ 2 } \)) such that it collides with the room at (\(\frac { 5 }{ 13 } ,\frac { 3 }{ 13 } ,1\)) and bounces of the walls repeatedly till it strikes one of the corners of room. Find the number of collisions the ball made with walls of the room before the ball finally strikes a corner.

**Details and assumptions**

1) The coordinates of the corners of the room are (0,0,0) ,(1,0,0) ,(0,1,0) ,(0,0,1) ,(0,1,1) ,(1,0,1) ,(1,1,0) ,(1,1,1).

2) The room is **gravity free**.

3) All collisions with the walls of the room are elastic.

4) The ball and corners of the room are point sized i.e. ball must strike the corner exactly.No near cases must be considered.

5) While counting the number of collisions also count the first collision i.e. collision at (\(\frac { 5 }{ 13 } ,\frac { 3 }{ 13 } ,1\))

###### This problem is original and is inspired by This problem.

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