# Browsing category puzzles

## A Puzzle Full of Nines

A nice puzzle by way of @benjaminleis: This AIME problem is fun: pic.twitter.com/DhbviTqnqr — Benjamin Leis (@benjamin_leis) February 2, 2020 In case you can’t read that, we need to find the sum of the digits in $N = 9 + 99 + 999 + 999\dots999$, where the last number consists

## Captain Holt’s Seesaw

When Jake’s father (Bradley Whitford) comes to town, Jake is excited to see him, but Charles is wary of his intentions; Holt challenges Amy, Terry, Gina and Rosa with a brain teaser in exchange for Beyonce tickets. Brooklyn 99, S02 E18, Captain Peralta “Surely it isn’t that hard?” I have

## A Sudoku Surprise

Once upon a time, I was a sudoku fiend. They provided an outlet, a distraction, a hiding place – I could bury my head in one for, say, half an hour and whatever had been troubling me before was somehow less of an issue. It’s not an effective strategy for

## Cav, Catriona and some hexagons

It’s always fun to tackle a puzzle from one of Cav’s posts - in this case, a Catriona Shearer puzzle; it looks like my solutions are completely unrelated to his, although in reality I tend to sneak the odd peek and take some inspiration1 I also liked that Cav shared

## 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