The puzzles involved cubic numbers. Easy: Find a non-zero number expressible as a sum of two positive cubes in two different ways. The desired answer is 1729 = 12^3 + 1^3 = 9^3 + 10^3, the order two taxicab number.
Moderate: Find a non-zero number expressible as a sum of two cubes in three different ways. The desired answer is 728 = -10^3+12^3 = -1^3+9^3 = 6^3+8^3, the order three cabtaxi number.
Too hard: Find four different non-zero numbers so that the sum of any pair is a cubic number. The desired answer is (-4868, -3132, 4860, 8964). In pairs, the sums are the cubes of (-20, 24, -2, 18, 12, 16). Divided by two, that's (-10, 12, -1, 9, 6, 8), the same cubes as in the problem above. A third order Diophantine equation is needed to solve it.