Puzzle solutions for you. http://joshmaxsilverman.github.io 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... Mon, 22 Jun 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-06-22-can-you-fix-the-random-number-generator/ http://joshmaxsilverman.github.io/2026-06-22-can-you-fix-the-random-number-generator/ 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... Mon, 01 Jun 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-06-01-can-you-spot-the-sheep/ http://joshmaxsilverman.github.io/2026-06-01-can-you-spot-the-sheep/ 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... Mon, 18 May 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-05-18-fiddler-hide-and-seek/ http://joshmaxsilverman.github.io/2026-05-18-fiddler-hide-and-seek/ 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... Mon, 11 May 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-05-11-fiddler-randomade/ http://joshmaxsilverman.github.io/2026-05-11-fiddler-randomade/ 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.... Mon, 20 Apr 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-04-20-have-you-heard-the-buzz/ http://joshmaxsilverman.github.io/2026-04-20-have-you-heard-the-buzz/ 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... Mon, 30 Mar 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-03-30-can-you-pile-the-primes/ http://joshmaxsilverman.github.io/2026-03-30-can-you-pile-the-primes/ 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... Mon, 23 Mar 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-03-23-can-you-trace-the-locus/ http://joshmaxsilverman.github.io/2026-03-23-can-you-trace-the-locus/ 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... Sun, 15 Mar 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-03-15-fiddler-cantor-distance/ http://joshmaxsilverman.github.io/2026-03-15-fiddler-cantor-distance/ 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... Tue, 10 Feb 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-02-10-writeup-curvature/ http://joshmaxsilverman.github.io/2026-02-10-writeup-curvature/ 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... Fri, 06 Feb 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-02-06-fiddler-all-star-winning-streak/ http://joshmaxsilverman.github.io/2026-02-06-fiddler-all-star-winning-streak/ 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... Wed, 04 Feb 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-02-04-fiddler-hexagonal-grid/ http://joshmaxsilverman.github.io/2026-02-04-fiddler-hexagonal-grid/ 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... Mon, 26 Jan 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-01-26-fiddler-bingo/ http://joshmaxsilverman.github.io/2026-01-26-fiddler-bingo/ 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... Thu, 01 Jan 2026 00:00:00 +0000 http://joshmaxsilverman.github.io/2026-01-01-JS-robot-javelin/ http://joshmaxsilverman.github.io/2026-01-01-JS-robot-javelin/ 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) =... Sat, 27 Dec 2025 00:00:00 +0000 http://joshmaxsilverman.github.io/2025-12-27-physics-covariant-derivative/ http://joshmaxsilverman.github.io/2025-12-27-physics-covariant-derivative/ 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 $\text{FPT}$ distribution.... Wed, 10 Dec 2025 00:00:00 +0000 http://joshmaxsilverman.github.io/2025-12-10-physics-diffusion-FPT/ http://joshmaxsilverman.github.io/2025-12-10-physics-diffusion-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... Mon, 08 Dec 2025 00:00:00 +0000 http://joshmaxsilverman.github.io/2025-12-08-fiddler-apollonian-dartboard/ http://joshmaxsilverman.github.io/2025-12-08-fiddler-apollonian-dartboard/ 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... Sun, 16 Nov 2025 00:00:00 +0000 http://joshmaxsilverman.github.io/2025-11-16-fiddler-garden-hose/ http://joshmaxsilverman.github.io/2025-11-16-fiddler-garden-hose/ How should you move in order to be unpredictable? - Question: You are a producer on a game show hosted by Randy “Random” Hall (no relation to Monty Hall). The show has three doors labeled 1 through 3 from left to right, and behind them are various prizes. Contestants pick one of the three doors at which to start, and... Sat, 08 Nov 2025 00:00:00 +0000 http://joshmaxsilverman.github.io/2025-11-08-fiddler-randy-hall/ http://joshmaxsilverman.github.io/2025-11-08-fiddler-randy-hall/ How common can league office scheming rig full counts to be? - Question: The Artificial Automaton Athletics Association (Quad-A) is at it again, to compete with postseason baseball they are developing a Robot Baseball competition. Games are composed of a series of independent at-bats in which the batter is trying to maximize expected score and the pitcher is trying to minimize expected... Sat, 01 Nov 2025 00:00:00 +0000 http://joshmaxsilverman.github.io/2025-11-01-JS-robot-baseball/ http://joshmaxsilverman.github.io/2025-11-01-JS-robot-baseball/ Where will you be when you can't stop winning? - Question: You’re playing a game of “high-low,” which proceeds as follows: First, you are presented with a random number, $x_1$, which is between $0$ and $1$. A new number, $x_2$, is about to be randomly selected between $0$ and $1$, independent of the first number. But before it’s selected, you... Sun, 07 Sep 2025 00:00:00 +0000 http://joshmaxsilverman.github.io/2025-09-07-fiddler-high-low/ http://joshmaxsilverman.github.io/2025-09-07-fiddler-high-low/