In the past decade computer science literature has studied the
effect of introducing random choices to classical processes. For example,
sequentially place n balls into n bins. For each ball, two bins are sampled
uniformly and the ball is placed in the emptier of the two. This process
does a much better job of evenly distributing the balls than the
"choiceless" version where one places each ball uniformly.
Consider the continuous version: Form a random sequence in the unit interval by having the n*th* term be whichever of two uniformly placed points falls in the larger gap between the previous n-1 points. I'll confirm the intuition that this sequence is a.s. equidistributed, solving an open problem from Itai Benjamini, Pascal Maillard and Elliot Paquette. The history goes back a century to Weyl and more recently to Kakutani. Several open problems will be discussed.