Fascinating. (I'm not familiar with the equidistribution of pi digits in record-calculations, though.) I am somewhat familiar with the arcsine law for penny-matching, which implies that in the limit as the number N of coin flips approaches oo, the fraction of the N flips that are (say) heads that is *least* likely is 1/2 — just the opposite of what I would have guessed before taking a probability course in college that used Feller vol. 1 as the text. Now I'm wondering about something similar for a 3-sided coin. —Dan ----- Everyone is acquainted with record-calculations of pi and how their (base ten) digits are equally distributed. Closely, but not of course exactly. By contrast, in base two there are seemingly an infinite number of initial digits of pi that contain exactly the same number of zeros and ones: https://oeis.org/A039624 What a difference adding one digit/dimension can make. In base three, the random walk of digits allows the initial 15 and 18 digits of pi to be exactly equidistributed by sheer small-numbers chance [in base four, the first 4 digits (3.021); in base five, the first 75 digits]. I thought I had a shot at finding a third solution in base three but it was not to be: http://gladhoboexpress.blogspot.ca/2016/11/a-deep-walk-in-base-three-pi.html -----