Indexing the archive…
Your Universe of Digital Possibilities
A fifty-slot album and a random dispenser: the first half fills in a sprint, the last three slots eat the evening. Every coupon is equally likely, forever — and the wait still climbs, because the target shrinks while the dice don’t. Count every duplicate and watch the harmonic staircase assemble from your own bad luck.
Drawing uniformly from N equally likely coupons, the k-th new one costs N/(N − k + 1) draws on average — one at the start, N at the end. The total is N times the harmonic number: the last coupon alone costs as much as the first N/2 together.
Completion times are not bell-shaped: centred at N ln N with width N, they follow the double exponential of extreme values, with a long right tail. A thousand albums finish over a spread of several N, and one is always out near twice the mean.
The draws needed until every coupon has appeared m times — N ln N + (m − 1)·N ln ln N + O(N). The second complete set costs only the ln ln term, which is why two collectors who swap duplicates each finish at about two thirds of the solo cost.
The Collector is the edition’s “when” for finishing: coverage, collections, the last flaky test. The probabilistic substrate is The Walk’s; the information reading is The Code’s — a duplicate is a draw that carries zero bits. In the edition, The Queue shares the “the crowd is the data” rendering, and the finale, The Proton, inverts the trick: there, more collectors buy less wait.