The fraction that maximises long-run growth is exactly the edge, 2p-1, which is 0.2 on this coin, and one derivative gets you there. Double it and the growth rate is -0.0024469 a flip, negative on a game that leans your way three hundred times in a row, and the crossing happens at 0.3894 rather than at 0.4. The article carries the exact median over 300 flips, 25 dollars to 10504.19 at the optimum and to 12.00 at double, the reason about 48 percent of overbettors still finish ahead anyway, and the place where the textbook approximation mean minus half the variance returns the opposite sign.
The midpoint of p and q sits strictly between them, and consecutive means precisely that no prime lives in that interval, so the answer is never and the proof is two lines with no arithmetic in it. The pair 2 and 3 survives for a different reason, since five halves is not an integer, and it is the only such pair.
Halving the length and timing the flame is not a biased estimator of half the time, it is unrelated to it: across four thousand random cords the midpoint method scattered from under fifteen seconds to over forty-five. Lighting both ends gives exactly thirty on every cord, by an argument that never evaluates the burn rate.
Fifty white marbles, fifty black, two jars, and a fair coin choosing which jar gets drawn from. The even split gives exactly one half, and so does every other split where the jars are the same size. Isolating a single white marble reaches 74/99, an exchange argument proves nothing beats it, and three quarters turns out to be a ceiling no arrangement ever touches.