Three Children, One Coin, and Eight Thirds of a Flip
A fair coin cuts probabilities into halves and quarters, and a short argument about the prime factorisation of two shows it can never reach one third in a bounded number of flips. Dropping the bound fixes it: flip twice, bin the tail-tail, and each child holds exactly a third for 8/3 flips on average. That naive scheme turns out to be the best any coin-flipping procedure can do for three outcomes, which stops being true at five.