Movie Critics

You run a film magazine, Groovy Movies, and you have been invited to attend a new film festival. The festival organizers will screen 30 films evenly distributed across three different screens. Each film will premiere at this festival, and you want to get the scoop on which one was the best. The problem is, though, that because there are three screens, you don’t know which screen will show the best film. You could watch only Screen A, see the best movie there and report on it, but it may not be as good as one of the movies on Screen B or C. Some more details you know from your many years of experience in the magazine biz: Whenever a film is playing on one screen, the other two screens also have a film playing. But there is enough time between each movie that one person can always watch the nth round on one screen and the n+1th round on another screen. The best movie on one screen will never play at the same time as the best movie on another screen. However, you don’t know what time slots they will occupy for each theater. All of your reviewers are good rankers — they won’t have any disagreement about which movies are better than others that they’ve seen. (They’re ordinal reviewers, in other words.) That said, all of your reviewers are terrible raters, so they cannot give an objective measure of how good a movie was (a 9 out of 10, say) and compare it to another reviewer’s measure of how good a different movie was. (To put it another way: They aren’t cardinal reviewers.) With all that in mind, if you want to know for sure what the best film at the festival was, what is the minimum number of reviewers you would need to send to the festival? Extra credit: What if there were more movies shown per screen? What if there were more screens? [Solution]

Thieving Houses

A town of 1,000 households has a strange law intended to prevent wealth-hoarding. On January 1 of every year, each household robs one other household, selected at random, moving all of that house’s money into their own house. The order in which the robberies take place is also random and is determined by a lottery. (Note that if House A robs House B first, and then C robs A, the houses of A and B would each be empty and C would have acquired the resources of both A and B.) Two questions about this fateful day: What is the probability that a house is not robbed over the course of the day? Suppose that every house has the same amount of cash to begin with — say $100. Which position in the lottery has the most expected cash at the end of the day, and what is that amount? [Solution]

Long Division

In the long division below, each asterisk represents a whole number — any digit from 0 to 9. Reconstruct all the calculations, given that there is no remainder. [Solution]

Lucky Derby

The bugle sounds, and 20 horses make their way to the starting gate for the first annual Lucky Derby. These horses, all trained at the mysterious Riddler Stables, are special. Each second, every Riddler-trained horse takes one step. Each step is exactly one meter long. But what these horses exhibit in precision, they lack in sense of direction. Most of the time, their steps are forward (toward the finish line) but the rest of the time they are backward (away from the finish line). As an avid fan of the Lucky Derby, you’ve done exhaustive research on these 20 competitors. You know that Horse One goes forward 52 percent of the time, Horse Two 54 percent of the time, Horse Three 56 percent, and so on, up to the favorite filly, Horse Twenty, who steps forward 90 percent of the time. The horses’ steps are taken independently of one another, and the finish line is 200 meters from the starting gate. Handicap this race and place your bets! In other words, what are the odds (a percentage is fine) that each horse wins? [Solution]

Paint Balls

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? [Solution]