Challenge problems

Problem 1

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off. What is the minimum number of times the lights must be turned off so that everyone will have left the room?

Problem 2
Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square. What is the probability that none of the cats collide? Challenge Problem 2 Solution

Problem 3

You working in a nuclear power plant and have 100 switches to check. At the start of the day, each switch is turned off. You make a first pass, checking each one as you go, and if the switch is off, you flip it on; if it is on, you flip it off. After all 100 switches have been checked, you make another pass, now only visiting every 2nd switch and proceed in the same way, i.e. if the switch is on, you flip it off and vice versa. Again, after all 100 switches have been checked, you pass again, now visiting every 3rd switch, continuing on in the same manner as before. This continues until you only have to visit the 100th switch. Now, which switches will be turned on and which will be off?