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]

Lever Box

Shut the Box is a traditional pub game played with dice and a special box. The rules are as follows: The game’s box features nine tiles bearing the numbers 1 through 9. Those tiles rest on flippable levers, all of which begin the game “open.” To start the game, you roll two standard dice. You can then “shut” any combination of number tiles that add up to the total of your dice. For example, if you roll 8, you could shut the 8; or the 1 and 7; or the 1, 3 and 4; and so on. Once a number is shut, it stays shut. After you’ve shut your chosen numbers, you roll again and repeat the process. (During the game, once the 7, 8 and 9 are shut, you may choose whether to roll one die or two dice. If any of those numbers are still open, you must roll two dice.) You win if you close all of the numbers before you run out of legal moves — that is, if you can’t shut numbers that add up to your dice count. If you play perfectly, what are your odds of successfully “shutting the box”? (fivethirtyeight) [Solution]

Shifted Double

Imagine taking a number and moving its last digit to the front. For example, 1,234 would become 4,123. What is the smallest positive integer such that when you do this, the result is exactly double the original number? (For bonus points, solve this one without a computer.) [Solution]