Puzzles.

How many people before a shared birthday is more likely than not?

How many boxes to complete the set? E[T] = n·Hₙ — surprisingly large.

Drop needles on lined paper to estimate π. Watch the estimate converge.

Two players bet until one goes bankrupt. Starting stake and bias determine the odds.

Reject the first 37%, hire the next best. Optimal stopping at 1/e ≈ 37%.

How likely that no hat returns to its owner? The exact probability approaches 1/e ≈ 36.8% as the number of items grows.

A gets a votes, B gets b. P(A strictly ahead throughout) = (a−b)/(a+b). Proved by reflection.

Balls through pegs. The binomial builds itself as a bell in real time.

Follow permutation cycles and explain why coordinated success is not independent.

One parity bit makes every answer after the first certain.

Track overlapping pattern states and distinguish first waiting times from frequency.

In n coin flips, how long is the longest streak? The answer grows like log₂ n.

Break a stick twice at random. The pieces form a triangle exactly 1/4 of the time.

Throw n balls into n bins. Max load grows like ln n / ln ln n.

Pick two bins at random, take the lesser-loaded. Max load collapses to ln ln n.

Measure how a random cut-and-interleave shuffle approaches uniform permutations.

Reinforcement changes future draws while the expected urn fraction stays fixed.

Moving one ball flips parity; allowing a hold changes convergence.

A non-standard pair of dice with the same sum distribution as 2d6.

Count what remains when a pocket is first discovered empty.

Reveal record highs and derive their harmonic expected count.

Compare each offer with the expected value of continuing.

Compare finite return probabilities with recurrence in one and two dimensions and transience in three.

Counting paths merge when they reach the same card; contact is not guaranteed.