**Probability**

The Problem of Points: Pascal and Fermat both put up 600 francs into a bet. They will flip a coin, and if heads comes up Fermat gets a point. If tails, Pascal gets a point. The first person to get 10 points wins and gets all 1200 francs. However, when Fermat is winning 8 to 7, he receives an urgent message to return to his hometown immediately and see his sick friend. The game is stopped, but how should be the 1200 francs be split?

Matching problem: If you randomly assign N names to N people (who each have a unique name), what is the probability that none of them are correctly named?

Matching birthdays: Fun experiment. How many students in the class do you need to see if at least two share the same birthday?

Buffon needle problem: A Monte Carlo method way of calculating pi.

Pass the beer: A counterintuitive stochastic process.

Two envelope paradox: A counterintuitive example brings up the idea of prior probabilities.

Monty Hall problem: Another thought exercise.

Deal or no deal: Risk aversion, and why expectations may not be the most accurate estimate of utility. Also see St. Petersburg Paradox.

Medical tests and Bayes rule: The usual introduction to conditional probability.

Betting on different beliefs: If two people have different beliefs about an outcome, you can make bets that guarantee you come out ahead. This also brings up the idea of prior probabilities (i.e. the gain comes from exploiting people who think their beliefs are certain when they could be wrong.).

**Calculus**

Cylinders in spheres: (From Kepler) Inscribe a cylinder of maximal volume inside a given sphere. (See Famous Puzzles of Great Mathematicians)

L’Hospital’s Pulley Problem: Let A and B be points on the ceiling that are 1 meter apart. From B, let there be a string of length r < 1 with a pulley at the end. From A, let there be a string of length l, such that l+r > 1, so that the string can go through the pulley to a mass. Assuming the strings and pulley are massless and frictionless, at what distance from the ceiling does the weight end up? (Hint: the mass ends up at its lowest point.)

The brachistochrone problem: Given two points A and B, construct the curve such that, if a mass is only acted on by gravity, the mass starts at rest at A and reaches B in the shortest amount of time.

Suspended Rope: What is the curve of a suspended string?

Beads and holes: Take two beads with a different radius. Drill a hole in each so that height of the remainder is the same. Which shape has the most volume left?

Surface area of sphere slices: Make equally thick horizontal slices of a sphere. The surface of each slice is the same.

A quick way to find square roots: Based off Newton’s method. Let y be the number you are trying to find the square root of. Starting with x_0, let x_{n+1} be the average of x_n and y / x_n. This actually converge pretty fast.

The black hole: The rotation of y(x) = 1/x, x > 0, about the x-axis has finite volume but infinite surface area. How can you paint it?

Circles and arcs: Consider a unit circle and an arc of length s in the first quadrant. Create the shape bounded by the arc and its projection onto the x-axis. Ditto the y-axis. The sum of the area of these two regions is always the same, no matter where the arc is located, and their sum is s.

Volume of a pyramid: Can probably adapt this to find the volume of a 3d pyramid with arbitrary base.

Fourier series: An alternative to Taylor/Maclaurin series.

Relationship between volume and surface area of spheres: Differential geometry.

Cantor set: For infinite series. The Cantor set is an uncountable set which has Lebesgue measure zero.

Thinned-out harmonic series: A weird way to thin out harmonic series so it converges.

How to integrate a bell-curve:

Weirstrass’ pathological function: Continuous everywhere, differentiable nowhere.

Does order of addition matter?:

**Complex numbers**

Chords of a circle: Take N equidistant points on the unit circle. Pick one of those points, then draw chords from it to all other points on the circle. For any N, the product of the lengths of those chords is N.

i-i: i to the i is a real number. Also, ‘aye’ means yes, and ‘aye aye’ means I have heard the order and will carry it out.

Pingback: Project-based learning « One, one, two, three, five…