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]

Coco Nuts

Seven pirates wash ashore on a deserted island after their ship sinks. In order to survive, they gather as many coconuts as they can find and throw them into a central pile. As the sun sets, they all go to sleep. One pirate wakes up in the middle of the night. Being the greedy person he is, this pirate decides to take some coconuts from the pile and hide them for himself. As he approaches the pile, though, he notices a monkey watching him. To keep the monkey quiet, the pirate tosses it one coconut from the pile. He then divides the rest of the pile into seven equally sized bunches and hides one of the bunches in the bushes. Finally, he recombines the remaining coconuts into a single pile and goes back to sleep. (Note that individual coconuts are very hard, and therefore indivisible.) Later that night, a second pirate wakes up with the same idea. She tosses the monkey one coconut from the central pile, divides the pile into seven bunches, hides her bunch, recombines the rest, and goes back to sleep. After that, a third pirate wakes up and does the same thing. Then a fourth. Then a fifth, and so on until all seven pirates have hidden a share of the coconuts. In the morning, the pirates look at the remaining central pile and notice that it has gotten quite small. They decide to split the pile into seven equal bunches and take one bunch each. (Note: The monkey does not get one this time.) If there were N coconuts in the pile originally, what is the smallest possible value of N? [Solution]

Traffic Lights

We are to walk on a rectilinear grid $N$ blocks north and $E$ blocks east, making $N$ and $E$ crossings of timed intersections. The intersections are not synchronized with each other, but there is a constant time period $T$ such that, for a given intersection, the north-south and east-west signals alternate $T/2$ intervals of displaying “GO” (we may start across) and “NO-GO” (we may not start across), along with the number of seconds until the next signal change. What should our strategy be, so as to have the lowest expected total time waiting at intersections? [Solution]