Consider a standard, two-player, 52-card game of War. If I start with just the four aces, and you start with all 48 other cards, randomly shuffled, what are your chances of winning?
[Solution]

## Foul Shots

You’re hanging out with some friends, shooting the breeze and talking sports. One of them brags to the group that he once made 17 free throws in a row after years of not having touched a basketball. You think the claim sounds unlikely, but plausible. Another friend scoffs, thinking it completely impossible. Let’s give your bragging friend the benefit of the doubt and say he’s a 70-percent free-throw shooter.
So, who’s right? What is the number of free throws that a 70-percent shooter would be expected to take before having a streak of 17 makes in a row? And what if his accuracy was a bit worse?
[Solution]

## Colored Hats

You and six friends are on a hit game show that works as follows: Each of you is randomly given a hat to wear that is either black or white. Each of you can see the colors of the hats that your friends are wearing but cannot see your own hat. Each of you has a decision to make. You can either attempt to guess your own hat color or pass. If at least one of you guesses correctly and none of you guess incorrectly then you win a fabulous, all-expenses-paid trip to see the next eclipse. If anyone guesses incorrectly or everyone passes, you all lose. No communication is possible during the game — you make your guesses or passes in separate soundproof rooms — but you are allowed to confer beforehand to develop a strategy.
What is your best strategy? What are your chances of winning?
Extra credit: What if instead of seven of you there are $2^N−1$?
[Solution]

## Packing Spheres

What’s the smallest tetrahedron into which we can pack four spheres?
[Solution]

## Converging Ratios

Take a look at this string of numbers:
333 2 333 2 333 2 33 2 333 2 333 2 333 2 33 2 333 2 333 2 …
At first it looks like someone fell asleep on a keyboard. But there’s an inner logic to the sequence:
Each digit refers to the number of consecutive 3s before a certain 2 appears. Specifically, the first digit refers to the number of consecutive 3s that appear before the first 2, the second digit refers to the number of 3s that appear consecutively before the second 2, and so on toward infinity.
The sequence never ends, but that won’t stop us from asking us questions about it. What is the ratio of 3s to 2s in the entire sequence?
[Solution]