This week’s Fiddler is about hopping back and forth. You are a frog in a pond with an infinite number of lily pads in a line, marked “1,” “2,” “3,” etc. You are currently on pad 2, and your goal is to make it to pad 1. From any given pad, there are specific probabilities … Continue reading "Can you hop to the lily pad?" The post Can you hop to the lily pad? first appeared on Book Proofs.| Book Proofs
This week’s Fiddler is about the number 2025, in celebration of (almost) New Years! First puzzle: What is the greatest number of distinct primes that add up to 2025? Second puzzle: How can you assign a set of 20 distinct prime numbers to the 20 vertices of a dodecahedron, so that the numbers on the … Continue reading "2025 puzzle" The post 2025 puzzle first appeared on Book Proofs.| Book Proofs
This week’s Fiddler is an optimization problem about fitting particles in a box. You have three particles inside a unit square that all repel one another. The energy between each pair of particles is $1/r$, where $r$ is the distance between them. To be clear, the particles can be anywhere inside the square or on … Continue reading "Particles in a box" The post Particles in a box first appeared on Book Proofs.| Book Proofs
This week's Fiddler is about rounding! You are presented with a bag of treats, which contains $n \geq 3$ peanut butter cups and some unknown quantity of candy corn kernels (with any amount being equally likely). You reach into the bag $k$ times, with $3 \leq k \leq n$, and pull out a candy at| Book Proofs
This week’s Fiddler is about rounding! Let $\text{round}(x)$ be the value of $x$ rounded to the nearest integer. Suppose $x_1,\dots,x_n$ are independent uniformly distributed random variables in $[0,1]$. Find the probability that \[ \text{round}(x_1+\cdots+x_n) = \text{round}(x_1)+\cdots+\text{round}(x_n) \] My solution: [Show Solution] Let’s call the probability we seek $p(n)$. The values of the $x_i$ determine what … Continue reading "Round, round, get a round" The post Round, round, ...| Book Proofs
This week’s Fiddler is a challenging counting problem. Consider the following array of 25 squares: You are filling the array with rectangles by repeating the following two steps: Select one of the 12 squares along the outer perimeter that has not yet been selected as part of a rectangle. Form the largest rectangle you can … Continue reading "Tiling a Tilted Square" The post Tiling a Tilted Square first appeared on Book Proofs.| Book Proofs
This week’s Fiddler is about a generalized notion of “radius”. For a circle with radius $r$, its area is $\pi r^2$ and its circumference is $2\pi r$. If you take the derivative of the area formula with respect to $r$, you get the circumference formula! Let’s define the term “differential radius.” The differential radius $r$ … Continue reading "When is a triangle like a circle?" The post When is a triangle like a circle? first appeared on Book Proofs.| Book Proofs
This week’s Fiddler is about tiling a square with smaller squares. Suppose you have infinitely many 3-by-3 cm tiles and infinitely many 5-by-5 cm tiles. You want to use some of these tiles to precisely cover a square whose side length is a whole number of centimeters. Tiles may not overlap, and they must completely … Continue reading "Tiling squares" The post Tiling squares first appeared on Book Proofs.| Book Proofs
This week’s Fiddler is based on “Showcase Showdown” on the game show “The Price is Right”. Suppose we have some number of players. Player A is the first to spin a giant wheel, which spits out a real number chosen randomly and uniformly between 0 and 1. All spins are independent of each other. After … Continue reading "Showcase Showdown" The post Showcase Showdown first appeared on Book Proofs.| Book Proofs
This week’s Fiddler is a puzzle about adding digits over and over again. For any positive, base-10 integer $n$, define $f(n)$ as the number of times you have to add up its digits until you get a one-digit number. For example, $f(23) = 1$ because $2+3 = 5$, a one-digit number. Meanwhile, $f(888) = 2$, … Continue reading "How many times can you add up the digits?" The post How many times can you add up the digits? first appeared on Book Proofs.| Book Proofs