Hector's Puzzled

If you break a stick in two places at random, forming three pieces, what is the probability of being able to form a triangle with the pieces? If you select three sticks, each of random length (between 0 and 1), what is the probability of being able to form a triangle with them? If you break a stick in two places at random, what is the probability of being able to form an acute triangle — where each angle is less than 90 degrees — with the pieces? If you select three sticks, each of random length (between 0 and 1), what is the probability of being able to form an acute triangle with the sticks? [Solution]

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]

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]

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]

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