Where will the seven dwarfs sleep tonight?

The following problem appeared in The Riddler. It’s an interesting recursive problem.

Each of the seven dwarfs sleeps in his own bed in a shared dormitory. Every night, they retire to bed one at a time, always in the same sequential order, with the youngest dwarf retiring first and the oldest retiring last. On a particular evening, the youngest dwarf is in a jolly mood. He decides not to go to his own bed but rather to choose one at random from among the other six beds. As each of the other dwarfs retires, he chooses his own bed if it is not occupied, and otherwise chooses another unoccupied bed at random.

  1. What is the probability that the oldest dwarf sleeps in his own bed?
  2. What is the expected number of dwarfs who do not sleep in their own beds?

Here is my solution.
[Show Solution]

Colorful balls puzzle

This Riddler puzzle about an interesting game involving picking colored balls out of a box. How long will the game last?

You play a game with four balls: One ball is red, one is blue, one is green and one is yellow. They are placed in a box. You draw a ball out of the box at random and note its color. Without replacing the first ball, you draw a second ball and then paint it to match the color of the first. Replace both balls, and repeat the process. The game ends when all four balls have become the same color. What is the expected number of turns to finish the game?

Extra credit: What if there are more balls and more colors?

Here is my solution to the first part (four balls):
[Show Solution]

Here is my solution to the general case with $N$ balls:
[Show Solution]

Rope timing

This Riddler problem is all about timing:

Suppose you have four ropes and a lighter. Each rope burns at a nonconstant rate but takes exactly one hour to burn completely from one end to the other. You can only light the ropes at either of their ends but can decide when to light each end as you see fit. If you’re strategic in how you burn the ropes, how many specific lengths of time can you measure? (For example, if you had one rope, you could measure two lengths of time: one hour, by simply burning the entire rope from one end, and half an hour, by burning the rope from both ends and marking when the flames meet.)

Extra credit: What if you had N ropes?

Here is my solution:
[Show Solution]

Note: my solution is incomplete (see comments below!)

Microorganism multiplication

This Riddler is about microorganisms multiplying. Will they thrive or will the species go extinct?

At the beginning of time, there is a single microorganism. Each day, every member of this species either splits into two copies of itself or dies. If the probability of multiplication is p, what are the chances that this species goes extinct?

Here is my solution:
[Show Solution]

Here is a more technical (and more correct!) solution adapted from a comment by Bojan.
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Impromptu gambling with dice

This Riddler puzzle is an impromptu gambling game about rolling dice.

You and I stumble across a 100-sided die in our local game shop. We know we need to have this die — there is no question about it — but we’re not quite sure what to do with it. So we devise a simple game: We keep rolling our new purchase until one roll shows a number smaller than the one before. Suppose I give you a dollar every time you roll. How much money do you expect to win?

Extra credit: What happens to the amount of money as the number of sides increases?

Here is my solution:
[Show Solution]

For a more in-depth analysis of the distribution, read on:
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Fighting stormtroopers

This Riddler puzzle is about fighting a group of stormtroopers. Why are they so inaccurate anyway?

In Star Wars battles, sometimes a severely outnumbered force emerges victorious through superior skill. You panic when you see a group of nine stormtroopers coming at you from very far away in the distance. Fortunately, they are notoriously inaccurate with their blaster fire, with only a 0.1 percent chance of hitting you with each of their shots. You and each stormtrooper fire blasters at the same rate, but you are $K$ times as likely to hit your target with each shot. Furthermore, the stormtroopers walk in a tight formation, and can therefore create a larger target area. Specifically, if there are $N$ stormtroopers left, your chance of hitting one of them is $(K\sqrt{N})/1000$. Each shot has an independent probability of hitting and immediately taking out its target. For approximately what value of $K$ is this a fair match between you and the stormtroopers (where you have 50 percent chance of blasting all of them)?

Here is my solution.
[Show Solution]

The lonesome king

This Riddler puzzle is about a random elimination game. Will someone remain at the end, or will everyone be eliminated?

In the first round, every subject simultaneously chooses a random other subject on the green. (It’s possible, of course, that some subjects will be chosen by more than one other subject.) Everybody chosen is eliminated. In each successive round, the subjects who are still in contention simultaneously choose a random remaining subject, and again everybody chosen is eliminated. If there is eventually exactly one subject remaining at the end of a round, he or she wins and heads straight to the castle for fêting. However, it’s also possible that everybody could be eliminated in the last round, in which case nobody wins and the king remains alone. If the kingdom has a population of 56,000 (not including the king), is it more likely that a prince or princess will be crowned or that nobody will win? How does the answer change for a kingdom of arbitrary size?

Here is my solution:
[Show Solution]

The deadly board game

This Riddler classic puzzle involves a combination of decision-making and probability.

While traveling in the Kingdom of Arbitraria, you are accused of a heinous crime. Arbitraria decides who’s guilty or innocent not through a court system, but a board game. It’s played on a simple board: a track with sequential spaces numbered from 0 to 1,000. The zero space is marked “start,” and your token is placed on it. You are handed a fair six-sided die and three coins. You are allowed to place the coins on three different (nonzero) spaces. Once placed, the coins may not be moved.

After placing the three coins, you roll the die and move your token forward the appropriate number of spaces. If, after moving the token, it lands on a space with a coin on it, you are freed. If not, you roll again and continue moving forward. If your token passes all three coins without landing on one, you are executed. On which three spaces should you place the coins to maximize your chances of survival?

Extra credit: Suppose there’s an additional rule that you cannot place the coins on adjacent spaces. What is the ideal placement now? What about the worst squares — where should you place your coins if you’re making a play for martyrdom?

Here is my solution:
[Show Solution]

Betting on the world series

This Riddler classic puzzle is about placing bets on baseball:

You are a gambler and a Cubs fan. The Cubs are competing in a seven-game series against the Red Sox — first to four games wins. Your bookie agrees to take any even-odds bets on any of the individual games. Can you construct a series of bets such that the guaranteed outcomes are: You win \$100 if the Cubs wins the series and lose \$100 if the Red Sox win it?

The challenge here is that we don’t know ahead of time when the series will end. It could end in a four-game blowout, or it could last the full seven games. How should we construct our bets so that the result is the same regardless of series length? Here is my solution:
[Show Solution]

Splitting a hundred dollar bill

This Riddler puzzle investigates a method for deciding who should get a $100 bill found on the ground. It leads to some interesting consequences…

You and four statistician colleagues find a \$100 bill on the floor of your department’s faculty lounge. None of you have change, so you agree to play a game of chance to divide the money probabilistically. The five of you sit around a table. The game is played in turns. Each turn, one of three things can happen, each with an equal probability: The bill can move one position to the left, one position to the right, or the game ends and the person with the bill in front of him or her wins the game. You have tenure and seniority, so the bill starts in front of you. What are the chances you win the money? What if there were N statisticians in the department?

Here is my solution to the first part, assuming five statisticians.
[Show Solution]

Here is my solution to the second part, assuming $N$ statisticians.
[Show Solution]

For the brave and curious, this next section explores connections between the problem and Fourier Transforms, complex analysis, and Chebyshev polynomials. Fair warning: advanced math!
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