possiblywrong

It's halftime, and you are the team's mascot, shooting free t-shirts into the crowd of spectators. You are standing at the edge of the circular field, next to the field-level seating, looking up at the rest of the stadium, which is shaped like a conical frustum, with seating rising away from you at angle theta to the horizontal, all the way to the nose-bleed section (like this).

Ignoring air resistance, at what angle should you aim your shirt gun to reach seats as high as possible?

Aron and Nora are standing on a flat plane. Aron walks 1/2 mile, then 1/4 mile, then 1/8 mile, etc. ... but before each "step," he flips a fair coin to determine whether to walk east or north. That is, on step n, he walks 1/2^(n) mile either east or north at random.

At the same time, Nora does the same thing, but having started a mile east and a mile north of Aron (that is, Aron starts at the origin and Nora starts at (1,1)), and randomly walking west or south at each step (instead of east or north as Aron does).

After an infinite number of steps, what is the expected distance between Aron's and Nora's limit positions?

The math department has 40 faculty, from which a committee of 10 members must be selected. The committee must be approved by unanimous consent of the entire department (including the committee members), where each professor will approve a committee if and only if it contains at least one member they want on the committee. (A professor may or may not want themselves on the committee.)

Every pair of professors has at least one professor in common that they both want on the committee. Is it possible to find a committee that the department will approve?

Starting with a bag containing a set of dominoes, is it possible to arrange them in a line?

More details: A domino is a 1x2 rectangular tile with each square "end" labeled with a non-negative integer. A "double-D" set of dominoes contains (D+1)(D+2)/2 dominoes, exactly one with each possible unordered pair of labels in the range 0 to D; the common "double-six" set of 28 dominoes is shown here.)

Is it possible to arrange the dominoes in a single line, end to end, so that the labels match on the adjacent ends of consecutive dominoes? For what "double-D" sets can this be done? (For

The "Big Top" circus tent is roughly circular, supported by N tent poles arranged in a circle. An ant walking on the ground wants to approach the tent from the outside, visit one of the tent poles, then walk directly to another tent pole, then directly to another, and so on, visiting each tent pole exactly once, before departing the area... without ever crossing its own (straight-line) paths under the tent.

How many ways can the ant do this?

Consider the relation < on the set of ordered pairs (u,d) of non-negative integers, defined by (u_1,d_1)<(u_2,d_2) if P(X_1>Y_1)<1/2, where X_k~Beta(u_k+1,d_k+1). Can this relation be extended to a (strict) total order in a "natural" way?

For context and motivation, each comment in a Reddit thread has a number (u,d) of upvotes and downvotes, respectively. If we imagine votes on a comment as observed Bernoulli trials with unknown probability P(upvote)=p (and P(downvote)=1-p), then the "best" ranking algorithm sorts comments by the lower bound of the 80% Wilson score confidence interval estimate for p. (At least, that's [how it used to work](https://github.com/reddit-archive/reddi

Shuffle a deck of r=26 red cards and b=26 black cards. Starting with one dollar, at each turn wager any fraction 0<=f<=1 of your current bankroll, betting on the color of the next card dealt. Continue until the deck is depleted.

What is the maximum expected return from playing this game?

Bonus: is there a betting strategy that guarantees this return?

(This is not a new problem, but I've now seen it three times in recent weeks, so thought this group might find it interesting.)

I have a bottle of n pills, that I am supposed to take over the course of 2n days, taking just half of a single pill per day.

Each morning, I give the bottle a shake until, well, something comes out. (More on this later.) If it's a whole pill, I break it in half, take one half with a glass of water, and put the other half back in the bottle. If it's a half of an already-broken pill, I take it.

Question 1: What is the expected number E(X) of half pills remaining in the bottle, on the morning after I have removed the last whole pill from the bottle (having taken half and put the other half back in the bottle, so that, for example, E(X)=1 when n=1)?

Question 2: What is the probability of s

Motivated by this recent thread:

You are playing golf on the moon. What is the optimal loft angle of the club to maximize the distance that your ball travels?

Puzzle-style assumptions: the moon is a flat plane with a uniform downward gravitational acceleration of g=1.6 m/s^(2), with no atmosphere. You swing the club so that the club face strikes the ball at the bottom of your swing, i.e. moving horizontally, at v=40 m/s. The club-ball collision is perfectly elastic, and the mass of the ball is negligible compared to the mass of the club.

You have