There's a spaceperson with a very bouncy ball and a rigid box in the form of a cube with one face missing. One day she throws the ball into the box and notices the ball bounces off each face exactly once before exiting through the missing face.
(The ball travels in a perfectly straight line, being unaffected by air resistance, spin or any other forces other than the reactions with the box. Also the ball bounces symmetrically such that the incoming angle is identical to the outgoing angle and again is unaffected by spin. Also, the box cannot be moved while the ball is in motion.)
How many different combinations are there of the order in which the ball can bounce off all five faces?
On returning to Earth our spaceperson notices that new combinations are possible.
(All conditions are the same except the ball is now affected by gravity.)
How many different combinations are there of the order in which the ball can bounce off all five faces now?
In Newtown Middle School, there was a school boy named Chris who was an absolute troublemaker. One of his many schemes against the school was drawing squares in permanent marker on all the walls in the school. Thus, when the school decided to make a punishment for Chris, they decided to do something involving squares.
The school made Chris create all possible unique Greco-Latin squares using A-D and 1-4. (A 4x4 Greco-Latin square using A-D and 1-4 is a special 4x4 square. Each cell of the square has exactly one letter of the four and one number of the four within it. The end result will have every letter and every number used once in each row, column, and main diagonal of the square.)
Chris is a very slow boy, and after several hours, he figured out all the possible Greco-Latin squares. How many squares did he find?
Random: 
Bonus: 
