## Duo Dominoes

Paula would like to re-tile the hallway with those nice domino-shaped tiles she discovered while redoing the kitchen. The tiles are 1×2, and the hallway is just wide enough for one of these tiles to fit horizontally. Paula knows she doesn’t have to tile the hallway all the way to …

## Pennies

Suppose you have 8 pennies, and you arrange 7 of them in a cluster like this: Now suppose you take the eighth penny and place it at the top: And then you roll it around the outside of the cluster, never letting it slide and always staying in contact with …

## Swans

A flock of  swans is sitting peacefully on a pond. Then, suddenly, one of the swans hears something in a bush at the water’s edge, and it takes off and flies away. Suppose that if  swans have flown away, the probability of another swan subsequently joining them is (so the …

## City Council

A certain city is entirely inhabited by knights and liars. Knights, consistently chivalrous, always tell the truth. Liars, on the contrary, never tell the truth. One can never tell whether a citizen is a knight or a liar just by their appearance. The city council, recently elected, is composed of …

## Cut Corners

Paula is re-doing the floor tiling in the kitchen. The space is shaped like an 8×8 square, except that one corner is missing to make room for the refrigerator and the opposite corner is missing to leave space for the dishwasher. Paula found a domino-shaped (1×2) tile she’d like to …

## Troll Trouble

You have been captured by an evil troll. The troll doesn’t have much to do besides capture people, so to amuse itself it’s made a bargain with you: you are to play a certain game with the troll, and if you win, you go free. The game starts with a …

## Leapfrog

Three frogs are sitting on the coordinate plane: one at (0, 0), one at (1, 0), and one at (0, 1). They play a game of leapfrog such that whenever one frog hops over another, the leapee is at the midpoint of where the leaper leaped from and where the …

## A Phony Proof

Private detective mathematician Harriet Hesterton was sitting calmly at her desk one day absently flipping through case files when her phone rang. “My name is Gus Spurious,” said the voice on the other end, “and I have a problem. See, I was playing with a bit of algebra the other …

## Salt and Pepper

Chef Charles is creating a couple of salt-and-pepper blends to simplify the process of seasoning a dish. He starts with two ordinary salt and pepper shakers, both the same size and both filled the same amount. To make the seasoning blends, Charles measures out one teaspoon of pepper from the …

## Light Bulbs

Rachel’s math club meets in a classroom at the end of a long hallway in her school. This long hallway is lit by a row of 100 light bulbs, each with a chain attached to it that toggles the light bulb on or off when pulled. One day, Rachel arrives …