Harvey the Spider is putting on his argyle socks and boots in the morning. Naturally, he has 8 socks and 8 boots to put on, and for every one of his 8 legs, he has to put the sock on before the boot. Let’s number Harvey’s feet 1 through 8, call the argyle sock for his $i\textrm{th}$ foot $A_i$, and call the boot for his $i\textrm{th}$ foot $B_i$. Then one valid order Harvey could use for putting on socks and boots is

Paula would like to re-tile the hallway with those nice domino-shaped tiles she discovered while redoing the kitchen. The tiles are 1x2, 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 the back of the house, since the carpet from the den can stretch across some portion of the hallway; she hasn’t decided exactly how long the tiling will be, though.

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 the cluster: Viewing the whole scenario from above, how many 360-degree rotations does the outer penny make during one trip around the central cluster?

A flock of $n$ 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 $k$ swans have flown away, the probability of another swan subsequently joining them is $1-\frac{1}{k+3}$ (so the more swans fly, the more likely it is that more swans fly also). What is the probability that the entire flock of $n$ swans flies away once the first one takes off?

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 twelve of the city’s most eminent inhabitants. One day they come together to a council meeting. They all recognize each other and know which members of the group are knights and which are liars.

Paula is re-doing the floor tiling in the kitchen. The space is shaped like an 8x8 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 (1x2) tile she’d like to use to tile the space: How many ways are there for her to tile the kitchen with this shape? The solution can be found here.

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 pile of 50 pebbles. Players take turns removing either 1 or 2 pebbles from the pile. The player to take away the last pebble from the pile loses the game.

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 leaper landed. For example, here are two valid leaps in the game: (0, 0) to (2, 0) and then (0, 1) to (4, -1).

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 day, and I proved something that just looks completely wrong. I might’ve just made a breakthrough that changes the face of mathematics, but I can’t figure out how my result could be true!

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 pepper shaker and adds it to the salt shaker, stirring to mix in the pepper evenly; he then measures out one teaspoon of this salt-and-pepper mix, adds it back to the pepper shaker, and stirs.