EnigmaThis game, as well as Weighing and The Chain, comes from an enigma which furnishes valuable mathematical information. The enigma is very percise and straightforward, specifying the number of coins and weighings, as well as the fault of the heavier coin. Specific To GeneralThe enigma specifies 9 coins and 3 weighings, but it could very well have been 3 coins and 1 weighing (for beginners) or 27 coins and 3 weighings (as a point of reference for the second, difficult version). In general, if "n" represents the number of weighings, the total number of coins will be 3n. Problem-solvingThis seems an appropriate place to introduce the two principal forms of problem-solving,     one which is logical and inevitable, with only one unique solution,          and another which is by trial-and-error, with several possible solutions.The first game, with its 9 coins, 2 weighings, and heavier faulty coin,     is an ideal example of a logical process with only one possible solution,          the "9" of the coins immediately suggesting "32",               with its subdivision by "3" twice, the only possible solution.On the other hand, the second game, with its 3 weighings, 12 coins,          and its complicated "lighter or heavier" faulty coin,     has 24 (2 x 12) possibilities within the 27 available, leaving 3 possibilities unused,          and inevitably imposing trial-and-error with its multiple solutions.     The Diagonal Game at the end of Hop-Over certainly belongs in the trial-and-error category.