4 entries tagged instruments, newest first.
A hundred candidates arrive one at a time, you can only compare them to the ones you have already seen, and every rejection is final. The famous answer is to let 37 go by and then take the first one better than all of them, which lands the very best of the hundred 37 per cent of the time. It also finishes twentieth on average, because in the rounds where no record turns up — 37.4 per cent of them, a different number from the one above and close to it for the same reason — it is left holding whoever walked in last. A different rule, which relaxes what it will accept as the queue runs out, takes the best only 26 per cent of the time and averages rank 3.6 — and never worse than 3.8695 however long the queue.
We each name a run of three coin flips, then flip a fair coin until one of the runs appears. You name yours first — and that is the losing move. Whatever you pick there is an answer that beats it: HHH loses to THH seven times in eight, and the four safest openings still lose two times in three. This page works out all fifty-six ordered pairs in exact fractions by Conway's leading numbers, solves the same fifty-six again by clearing an absorbing Markov chain over rationals, plays two thousand seeded rounds against every opening, and draws the four-sequence cycle that shows why no best sequence exists.
Four thousand drivers, two routes, sixty-five minutes each. Open a new road between the two halves that takes no time at all to drive, and every one of them takes eighty. Nobody is behaving badly and nothing is congested by accident: the eighty-minute outcome is the only stable one. This page works the whole thing out in exact fractions, shows that the harm exists only between three thousand drivers and nine thousand, peaks at exactly 22.5 minutes, and lands on the same four-thirds that bounds every network of this kind.
In February 1944 the Allies estimated German tank production two ways. The spies said 1,400 a month. The serial numbers on captured gearboxes said 270. The German records, found after the war, said 276. The whole method is one line — the highest number you have seen, plus one more average gap — and this page shows that it is exactly unbiased, that the obvious alternative is more than twice as wide, and how many serial numbers you need before the answer is worth printing.