Puzzle #27: Hearts

In a four-player game of Hearts, the sum of the scores after a round is divisible by 26. In some cases, all of the scores are cubes.

A player’s score can range from 0 to 125 points. After each round, at least one player’s score must be less than 100 points (since the game ends when the first player reaches 100 or more points). Other than that, do not concern yourself with whether a solution is actually possible under the scoring rules of Hearts.

With the constraints of the previous paragraph, what are the possible sets of scores? Do not report all of the permutations of a solution, just the one in order from lowest to highest score.

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, December 11, 2013 at noon Pacific Time. I will post the answer shortly after.