Math Riddles

Question: You have three coins. One always comes up heads, one always comes up tails, and one is just a regular coin (has equal change of heads or tails). If you pick one of the coins randomly and flip it twice and get heads twice, what is the chance of flipping heads again?

Question: You're going to a friends house and want to give them some brownies. On the way to their house you cross 5 bridges. At each bridge you must pay a toll of half of your brownies to the owner. But they are all very nice and give you back a brownie from what you give them.

How many brownies must you start with to give your friend 2 brownies?

Question: You have a flashlight that takes 2 working batteries. You have 8 batteries but only 4 of them work.

What is the fewest number of pairs you need to test to guarantee you can get the flashlight on?

Question: You create your own Where's Waldo game on a piece of paper. You want your friend to find Waldo but they think you might be pulling a prank on them by not putting Waldo on the paper at all.

How can you prove to your friend that Waldo is on the paper without ruining it for them?

Question: A prison has 23 prisoners in 23 different cells. The prisoners have no way to communicate with each other in any way from their cells. There is another room, the rec room, that has two switches on the wall (A and B). The switches have on and off positions but they start in an unknown position.

Prisoners are randomly taken to and from the rec room one at a time. They must change the position of only one of the two switches each time they go to the room. At any point a prisoner can yell out, "Every prisoner has been here!" If the prisoner is correct that all of the prisoners have visited the rec room, then they all go free. If they aren't correct then they are all executed.

Before they start they are given one planning session during which they can discuss a method to win the game. What method can they use to ensure they all go free?