Consider the following two player game:

Two players take turns naming numbers 1 through 9. Players may not name a number that has already been named. If at any point three of the numbers that one of the players has named add up to exactly 15, that player wins. After all 9 numbers have been named, if neither player has already won, the game is declared a tie.

Is this game a first player win, a second player win, or a tie? This problem shouldnt be too difficult to solve on a computer, but you get bonus points if you can convince a friend of the correct answer in two minutes with out a computer. Solution in comments.

Imagine that you have a collection of very weird dice. For every prime between 1 and 1000, you have a fair die with that many sides. Your goal is to generate a uniform random integer from 1 to 1001 inclusive.

For example, using only the 2 sided die, you can roll it 10 times to get a number from 1 to 1024. If this result is less than or equal to 1001, take that as your result. Otherwise, start over.

This algorithm uses on average 10240/1001=10.228770… rolls. What is the fewest expected number of die rolls needed to complete this task?

When you know the right answer, you will probably be able to prove it. This puzzle is my own design. Solution in the comments.

Imagine the following two player game. Alice secretly fills 3 rooms with apples. She has an infinite supply of apples and infinitely large rooms, so each room can have any non-negative integer number of apples. She must put a different number of apples in each room. Bob will then open the doors to the rooms in any order he chooses. After opening each door and counting the apples, but before he opens the next door, Bob must accept or reject that room. Bob must accept exactly two rooms and reject exactly one room. Bob loves apples, but hates regret. Bob wins the game if the total number of apples in the two rooms he accepts is a large as possible. Equivalently, Bob wins if the single room he rejects has the fewest apples. Alice wins if Bob loses.

Which of the two players has the advantage in this game?

This puzzle is my own design, and it is a lot more interesting than it looks at first. I will post the solution in the comments.