Hector's Puzzled

Say you have an “L” shape formed by two rectangles touching each other. These two rectangles could have any dimensions and they don’t have to be equal to each other in any way. Using only a straightedge and a pencil (no rulers, protractors or compasses), how can you draw a single straight line that cuts the L into exactly two pieces of exactly equal area, no matter what the dimensions of the L are? You can draw as many lines as you want to get to the solution, but the bisector itself can only be one single straight line. [Solution]

You know the following things about the delicate ecology of scorpions and meerkats: If there were no meerkats, the population of the scorpions would double every month. 2.If there were no scorpions, the population of the meerkats would halve every month. 3.If you had exactly 20 scorpions and five meerkats, both populations would not change at all. What is the number of meerkats when the desert has as many scorpions as possible? What is the number of meerkats and the number of scorpions when the meerkat population is increasing by four meerkats per month and the scorpion population is decreasing by two scorpions per month? These questions rely on the Lotka-Volterra model of predators and prey. That model makes certain assumptions, including that the prey (the scorpions here) always find ample food, the predators (the meerkats) have limitless appetite, and the rate of change of a population is proportional to its size. Extra credit: If you start with 100 scorpions and 10 meerkats, what is the maximum number of meerkats you can have and how long would it take for both populations to return to their starting states? [Solution]

I have a matching game app for my 4-year-old daughter. There are 10 different pairs of cards, each pair depicting the same animal. That makes 20 cards total, all arrayed face down. The goal is to match all the pairs. When you flip two cards up, if they match, they stay up, decreasing the number of unmatched cards and rewarding you with the corresponding animal sound. If they don’t match, they both flip back down. (Essentially like Concentration.) However, my 1-year-old son also likes to play the game, exclusively for its animal sounds. He has no ability to match cards intentionally — it’s all random. If he flips a pair of cards every second and it takes another second for them to either flip back over or to make the “matching” sound, how long should my daughter expect to have to wait before he finishes the game and it’s her turn again? [Solution]

The live smartphone game show HQ Trivia has taken the world by storm. In the game, you face a sequence of 12 multiple-choice trivia questions, each with three choices. If you answer all 12 correctly, you win a cash prize!2 Get one question wrong, however, and you are eliminated. If you didn’t know much trivia but did know strategy, how many phones would you need to guarantee that you’d win the cash on one of them? Cool extra credit: The real-life game has an added wrinkle: extra lives, which you can earn by referring others to the game. You can use an extra life after you get a question wrong, and you continue just as if you had gotten that question right. However, you can use only one of these per game per phone. Assuming that all your phones have an extra life, how many phones do you need to guarantee a victory now? [Solution]

Let’s say there are $N$ people going for a run, and assume that each person’s preferred running speed, call it $X_i$, is independent and normally distributed, with a mean of $\mu$ and a variance of $\sigma^2$. Or, mathematically, \[X_i \sim N(\mu,\sigma^2)\] Now, for any given person, let’s assume that the person has a running buddy if there’s another person in the group whose preferred speed is about the same as theirs. Specifically, we’ll say that person $i$ and person $j$ can be running buddies if their preferred speeds are within some number $s$ of each other — that is, if $|X_i - X_j| \leq s$. How large does $N$ need to be before the probability of each person having a running buddy is $99$ percent? (Assume $\mu$, $\sigma^2$ and $s$ are fixed.) [Solution]