Rug Rejection

A manufacturer, Riddler Rugs™, produces a random-pattern rug by sewing 1-inch-square pieces of fabric together. The final rugs are 100 inches by 100 inches, and the 1-inch pieces come in three colors: midnight green, silver, and white. The machine randomly picks a 1-inch fabric color for each piece of a rug. Because the manufacturer wants the rugs to look random, it rejects any rug that has a 4-by-4 block of squares that are all the same color. (Its customers don’t have a great sense of the law of large numbers, or of large rugs, for that matter.) What percentage of rugs would we expect Riddler Rugs™ to reject? How many colors should it use in the rug if it wants to manufacture a million rugs without rejecting any of them? [Solution]

L Divided

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]

Hungry Meerkats

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]

Matching Cards

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]

Running Buddies

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, 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]