Puzzle solutions for you.
When every team is the same, how best is the best? - Question: The Fiddler Baseball League consists of exactly two teams of equal skill: the Algebraists and the Geometers. Over the course of a season, these two teams play each other 162 times. Each team has an equal chance of winning each game, and the results of games are independent of...
How well can you win when you're the only one with a plan? - Question Congratulations to Fiddler Nation for making it to the semifinals of the World Cup! All four teams that made it this far are equally matched in that they each possess the same total amount of “energy.” In advance of each semifinal game, teams must independently decide how much of...
Two champion cyclists, each optimized for a different facet of their sport, will tie if you can stretch the hill just so. - Question: This time, we’ll be looking at a model for a cyclist’s speed $v$ as a function of their pedaling power $P$, their mass $m$, and the ground’s angle of inclination $\theta$: \[v=\frac{P}{m\sin{\theta}+10}\] In cycling, roads are marked with a gradient $g$, which is a…
Based on your friend's cryptic revelations, what do you expect is the biggest possible number? - Question I think the random number generator on my calculator might be malfunctioning. Oh no! Under normal conditions, it should generate random numbers between 0 and 1. But my suspicion is that the calculator is “tanked,” meaning it only generates random numbers between 0 and some value $0 < a...
What's the chance three randomly placed and oriented sheep can all see each other with their sheep eyes? - Question: Two sheep are at two random points inside a square pen. They are munching grass and staring in two random directions. Each sheep has a field of view that’s 180 degrees. What is the probability that they both see each other? Extra Credit: Now, three sheep are at three...
Minimize the time to find your hiding nephew, or don't, I'm not your dad. - Question: I am playing hide-and-seek with my nephew. I start at point O, whereas my nephew can hide at point A, B or C. I can walk from O to A in 2 minutes, from O to B in 3 minutes, from O to C in 4 minutes, and from...
How much will you pour using this deranged protocol? - Question: I’m preparing a mixture of “random-ade” using a large, empty pitcher and two 12-ounce glasses. First, I fill one glass with some amount of lemon juice chosen randomly and uniformly between 0 and 12 ounces. I fill the other glass with some amount of water, also chosen randomly and...
How big is big enough to buzz most of the time? - Question: I recently introduced my children to a game called “Buzz” (also known as “Fizz buzz”). In one particular variant of the game, anytime a number is a multiple of 7 or at least one of its digits is a 7, the player must say “buzz” instead of that number....
Can you send your six children home with the same weight of primes in their number bags? - Question: Suppose you want to make two groups with equal sums using the first $N_2$ prime numbers. What is the smallest value of $N_2$ for which you can do this? The answer is three! (Clearly, that wasn’t actually the puzzle.) The first three primes are $2,$ $3,$ and $5,$ and...
What area is mapped out by a string on some pegs? - Question: I have a loop of string whose total length is $10.$ I place it around a unit disk (i.e., with radius 1) and pull a point on the string away from the disk until the string is taut, as shown below. I drag this point around the disk in...
Random geometry from the Cantor distribution - Question: I start with a number line from 0 to 1, and then I remove the middle third. Then I take each of my remaining pieces (the segment from 0 to 1/3 and the segment from 2/3 to 1) and remove their middle thirds. Now I have four segments (from...
Where does it come from, how do we find it? - This note has a few angles on curvature. The first is to compute the curvature tensor, intuitively defined as the difference of repeated covariant derivatives in either order. The second is computing the tangent vector under parallel transport on an arbitrary line of latitude on the sphere. Third, we calculate...
What would be an impressive win streak in a hypothetical new all star game format? - Question: The Fiddler Basketball Association’s All-Star Game consists of two teams: “East” and “West.” Every year these two teams play a game, each with a $50$ percent chance of winning that’s independent of the outcomes of previous years. Many, many years into the future, you look at the most recent...
How long until a random walk steals 99% of your food, probably? - Question: Frankie has stored all of her food on lily pad A. However, her food has a tendency to “fly” away. Every second, the food that’s on every lily pad splits up into six equal portions that instantaneously relocate to the six neighboring pads. At zero seconds, all the food...
Ding ding ding, is it bingo time? - Question: A game of bingo typically consists of a $5$-by-$5$ grid with $25$ total squares. Each square (except for the center square) contains a number. When a square’s number is called, you place a marker on that square. The goal is to get “bingo,” which is five squares in a...
How should you counter the tosser who doesn't know you know they're cheating? - Question: It’s coming to the end of the year, which can only mean one thing: time for this year’s Robot Javelin finals! Whoa wait, you’ve never heard of Robot Javelin? Well then! Allow me to explain the rules: It’s head-to-head. Each of two robots makes their first throw, whose distance...
Where does it come from, how do we find it? - Question: How does the directional derivative generalize when basis vectors can vary? Argument We want to calculate the derivative of a vector $\mathbf{V}$ attached to a point that moves along the surface, in the coordinates of the surface, along some particular direction, at that point. Take a vector \[\mathbf{V}(X) =...
How long until a particle diffuses to $x$? - Question given a population of diffusing particles, what is the probability distribution of first passage times ($\text{FPT}(x,t)$) to position $x?$ Typically this is found by solving the diffusion differential equation to find the probability distribution, integrating that to find the survival, and then differentiating that to find the $\text{FPT}$…
How many points will you asymptotically approach? - Question: You are playing darts with your friend, Apollonius, who has brought his own dartboard. However, this dartboard is somewhat … different. Instead of being a circle divided into concentric rings and sectors, the dartboard is a unit circle (i.e., with radius $1$) that’s divided via an Apollonian gasket. In...
How far is the thirsty random tree from the gushing random hose? - Question: You and your assistant are planning to irrigate a vast circular garden, which has a radius of $1$ furlong. However, your assistant is somewhat lackadaisical when it comes to gardening. Their plan is to pick two random points on the circumference of the garden and run a hose straight...