the solutions in the monthly appear to strive for conciseness, which sometimes yield less insight. i also originally found it to be counterintuitive, in that it felt as if there should be a p <---> 1 - p symmetry. in fact, there is, if one also switches "below" and "above" (or "before" and "after"). here's my solution, which i think offers some insight. suppose the target ratio is a/b in lowest terms. keep track of an auxiliary score, which is defined as b * (number of successes) - a * (number of attempts) we are given that at some point, this score is negative, and at a later point it is positive. each successful attempt adds b - a to the score, and each miss subtracts a . in the original problem, a = 4 and b = 5 , so the score either increases by 1 or decreases by 4 . now it's easy to see why you must hit 0 exactly when going from below 0 to above. if the positive step b - a is greater than 1 this also suggests how to get from below 0 to above without actually hitting 0 , and indeed this is possible (because a and b are relatively prime). mike