Twenty-four possible answers against 3^3 = 27 outcome sequences leaves just enough room, and four against four splits those answers into exactly 8, 8 and 8. Six against six always tips, so it wastes the level outcome and leaves twelve answers for nine remaining sequences, which makes halving impossible rather than merely slow. The counting argument bounds outcome sequences rather than strategies, so it rules out every adaptive continuation at once, and the explicit schedule closes the positive half.
Take the centre, then mirror every move through it, and you place the last coin. The proof has three requirements and only one of them needs that opening move, which is the step a one-line answer skips. Central symmetry alone is not the condition: an annulus is centrally symmetric and the first player loses on it.
Three children's ages multiply to 36. Someone who knows the sum admits she cannot name them, and that admission is the only real clue in the problem. Eight triples, one repeated sum, and a second clue that eliminates nothing on its own yet decides everything once the first has run.
The pour back really was diluted, and the conclusion still does not follow: both jars finish at six cups, so whatever left one jar was replaced cup for cup by what arrived. That argument needs no fractions and survives terrible stirring, while the number 1.5 cups does not.
Taking one coin from each bag reads 9.9 ounces whichever bag is light, and the failure is blindness rather than imprecision. Loading i coins from bag i makes the dial an injective function of the culprit, at a cost of the tenth triangular number. A 45-coin variant is cheaper, and powers of two identify any subset of light bags from one reading.
Cut two diagonally opposite corners off a chessboard and 62 = 2 x 31 stays true, yet nothing fits. Writing the colour of a square as the sign (-1)^(i+j) turns the argument into arithmetic: every domino sums to zero, the two lost corners both carried +1, and the board left over is 30 against 32. The converse, Gomory's theorem, is the harder half and it goes the other way.