A practice trainer for the curse of dimensionality. Each round states how many dots fill a ball in p dimensions and the radius of a small window, then asks one of four things: the expected number of neighbors inside the window (N times r to the power p), the smallest dimension at which the window is expected to hold fewer than one dot, the number of dots needed to keep ten expected neighbors, or — by multiple choice, using a table of ball and cube volumes — the fraction of a cube that its inscribed ball occupies. Type a fraction (any equivalent form counts) or pick a choice; every answer gets a worked explanation, and each new round draws fresh numbers.

Dots spread evenly through a ball. A window’s share of the space is r × r × … × r — once per dimension. Count what that leaves you.