# Browsing category puzzles

## A puzzle from Sheena

A puzzle that came to me via @sheena2907: Choose two numbers, $x$ and $y$, uniformly from $[0,1]^2$. What’s the probability that $\frac{x}{y}$ rounds to an even number? What’s the probability that it rounds down to an even number? As always, spoilers below the line. Rounding One of my best approaches

## Bending a long bar

A nice thinker from Futility Closet: A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make? Feel free to

## Some puzzles from Cav

A couple of puzzles that came my way via @srcav today: Cav’s solutions to this one are here; mine are below the line further down. Interesting angle puzzle https://t.co/UN13XwwY3o pic.twitter.com/NyaQL0H7wE — Cav (@srcav) July 8, 2019 And to this one, here Have a go yourself before you read on! I’ve

## Powers and remainders

Over on Reddit, a couple of “last digit” puzzles crossed my path, and I thought I’d share the tricks I used, as much for my reference as anything else. 1) Show that the last digit of $6^k$ is 6, for any positive integer $k$. There’s a standard way to prove

## Barney’s triangles

A puzzle from @Barney_MT: Find angle BDC This turns out to be a bit more demanding than I expected. There are spoilers below the line, showing a solution that took rather more time and space than the final polished version does. Spoilers below the line! Adding in circles When I’ve

## A Trigonometric Puzzle

A puzzle that came to me via @realityminus3, who credits it to @manuelcj89: $\sin(A) + \sin(B) + \sin(C) = 0$ $\cos(A) + \cos(B) + \cos(C) = 0$ Find $\cos(A-B)$. There’s something pretty about that puzzle. Interestingly, my approach differed substantially from all of my Trusted And Respected Friends’. Spoilers below

## A Triangle In A Square

In a currently-recent (but by the time you read this, long in the past) Chalkdust1, @cshearer42 gave a puzzle that caught my eye. One of the things I love about Catriona’s puzzles is that you usually get two-for-the-price-of-one: there’s “getting the right answer”, which is not usually hard, and there’s

## Constructing the square root of 6

On Twitter, @RuedigerSimpson pointed me at an episode of My Favourite Theorem in which @FawnPNguyen mentioned a method for constructing $\sqrt{7}$: draw a circle of radius 4 construct a perpendicular to the radius at a distance of 3 from the centre the distance between the base of the perpendicular and

## A Harmonic Conundrum

This one came from user_1312 on reddit with a heading “This is a bit tricky… Enjoy!”. What else can we do but solve it? Let $m$ and $n$ be positive numbers such that $\frac{m}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{101}$. Prove that $m-n$ is a multiple

## Summing Products

Some days your mind wanders into an interesting puzzle: not necessarily because it’s a difficult puzzle, but because it has familiar result. Then the puzzle becomes, how are the two things linked? For example, I had cause to add up all of the numbers in the times tables - let’s