Probability

You can project n points into O(log n / eps^2) dimensions and keep every distance. The theorem underneath every vector database.

Constant space, no false negatives, a tunable false-positive rate, and the exact formula for the tradeoff.

Randomness probably doesn't add power; the surprise is that hardness is what buys you fake randomness.

What a hash function actually guarantees, as opposed to what it seems to do.

The length of the shortest program producing a string. Uncomputable, and the correct definition of "random" and "simple".

Random placement gives max load ~log n / log log n; picking the emptier of two random bins collapses it to log log n.

Averages of independent things are exponentially unlikely to stray.

Expectations add even when events are hopelessly dependent. Averaging as an existence proof: if the mean is 5, some input hits 5.

Collisions arrive at sqrt(N), not N. Sets every hash length, UUID size, and nonce width you will ever choose.