Hector's Puzzled

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]

Fans of Dungeons & Dragons will have fond feelings for four-sided dice, which are shaped like regular tetrahedrons. Some might have noticed, in those long hours of fantasy battle, that if you touch five of these pyramids face-to-face-to-face, they come agonizingly close to forming a closed pentagon. Alas, there remains a tiny angle of empty space left between two of the pyramids. What is the measure of that angle? [Solution]

There is a bathroom in your office building that has only one toilet. There is a small sign stuck to the outside of the door that you can slide from “Vacant” to “Occupied” so that no one else will try the door handle (theoretically) when you are inside. Unfortunately, people often forget to slide the sign to “Occupied” when entering, and they often forget to slide it to “Vacant” when exiting. Assume that 1/3 of bathroom users don’t notice the sign upon entering or exiting. Therefore, whatever the sign reads before their visit, it still reads the same thing during and after their visit. Another 1/3 of the users notice the sign upon entering and make sure that it says “Occupied” as they enter. However, they forget to slide it to “Vacant” when they exit. The remaining 1/3 of the users are very conscientious: They make sure the sign reads “Occupied” when they enter, and then they slide it to “Vacant” when they exit. Finally, assume that the bathroom is occupied exactly half of the time, all day, every day. Two questions about this workplace situation: If you go to the bathroom and see that the sign on the door reads “Occupied,” what is the probability that the bathroom is actually occupied? If the sign reads “Vacant,” what is the probability that the bathroom actually is vacant? [Solution]

There are two warlords: you and your archenemy, with whom you’re competing to conquer castles and collect the most victory points. Each of the 10 castles has its own strategic value for a would-be conqueror. Specifically, the castles are worth 1, 2, 3, … , 9 and 10 victory points. You and your enemy each have 100 soldiers to distribute between any of the 10 castles. Whoever sends more soldiers to a given castle conquers that castle and wins its victory points. (If you each send the same number of troops, you split the points.) Whoever ends up with the most points wins. But now, you have a spy! You know how many soldiers your archenemy will send to each castle. The bad news, though, is that you no longer have 100 soldiers — your army suffered some losses in a previous battle. What is the value of the spy? [Solution]

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]