Happy PI Day Problems
Imagine three pies. The pies are stacked on top of each other like a tiered wedding cake. The apple pie is biggest and on the bottom, there is a banana meringue pie in the middle, and a small cherry pie on top. Let’s play a “game” on these pies involving pennies. First, you flip 8 pennies (showing heads/tails with equal probability) and place them on the apple layer (evenlyspaced around the ring). There are no choices, and the “game” proceeds deterministically from this point. During each round: 

1) All of the pennies showing heads remain heads and advance by climbing up one layer (i.e. apple > banana, banana > cherry) moving inward along the straight line toward the center.
2) All of the pennies showing tails flip over to become heads, but do NOT advance.
3) After the movements are complete, any pennies that are overlapping with one another are removed from the game. (On the banana layer, a penny may overlap with each of its immediate neighbors – see diagram  and the cherry pie has only one penny slot, so if more than one penny reaches it during a round, they will overlap).
4) You win the game if there is a penny on the cherry pie at the end of a round. (You lose if this never happens – e.g. if all the pennies get removed from the game). What is the probability that you win (i.e. there was a round where precisely one penny advanced to the cherry pie level)?
Extra challenge: Change the game so that instead of removing overlapping pennies in step 3, you send them back to their original positions on the apple layer displaying tails. What is the probability of winning with these rules?
2015 Problem: A sphere of radius 1 is placed inside a cube of side length 4 such that the sphere is tangent to 3 of the cube faces. If another sphere of radius 1 is randomly placed in such a way that it is entirely contained within the cube, what is the probability that the two spheres intersect? 
2014 Problem: A hollow cylinder with a 6 meter radius contains two cylinders, which are each in contact with the walls of the outer cylinder. These cylinders have radii of one meter and two meters, respectively. Both cylinders rotate with the same angular velocity and remain in contact with the walls of the hollow cylinder without slipping. If at time zero both cylinders are on opposite ends of the diameter of the large cylinder and are both heading clockwise around the outer cylinder, how far in meters will the center of the small cylinder travel before the large cylinder collides with it? Express your answer with at least 4 significant digits. 
Forrest made a perfectly circular pie of radius 1. Susa randomly placed a perfectly circular ring of pineapple of radius 1/3 on top of the pie, such that none of it hung over the edge. If you slice the pie all the way across at a random angle through the center, what is the probability you will hit the piece of pineapple? (Please state your answer with three decimal points.) 2012 Problem: You have three pies of diameter 9 inches. Without cutting any of the pies a brilliant genie places two of the pies on a circular table, such that the center of mass of each pie is directly over the table. The genie agrees to grant your wish if you can place the third uncut pie ENTIRELY on the table such that it does not overlap with either of the first two pies. How big must the table’s diameter be, such that you are guaranteed to be able to earn your wish? 2011 Problem: As Gabby was both doing math problems and making a piday pie last night, she dropped her pencil onto the pie. Her pie is a perfect circle and her pencil is the same length as the radius of the pie. Assuming the pencil is lying flat on top of the pie and that the center of the pencil is over the pie itself (otherwise it wouldn't balance!), what is the probability that part of the pencil extends beyond the edge of the pie? (Express your answer in terms of π!) I think this one is harder than last year's problem, but it can be done without any real calculus, although a calculus concept is sort of required... 
If you randomly slice a perfectly circular pie 3 times, each slice along a diameter, thus making 6 pieces of pie that meet at the center, what is the probability that at least 1 piece of the pie is greater than one quarter of the entire pie? Hint: The answer is not 1/2 and it can be done without calculus, but calculus may make it easier.. Contact me if you want to check your answers. 