This week’s Fiddler is about the popular NY Times puzzle Letter Boxed. you must connect letters together around a square to spell out words (they don’t have to be actual English words!). However, from any given letter, the next letter cannot be on the same side of the square. How many distinct valid sequences are … Continue reading "Letter Boxed" The post Letter Boxed first appeared on Book Proofs.| Book Proofs
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