Equals Times

Exercise 10.3 - Three people own differently sized houses, whose exterior surface areas are A, B, and C Planck lengths squared. Of course, since Planck lengths are indivisible, their houses are obviously a positive integer number of Planck lengths squared. Anyway, these people all paint at constant rates (painting areas takes time proportional to their areas). As it so conveniently happens, they each take precisely 1 day to paint their own houses. Furthermore, when B and C together paint A's house, A and B together paint C's house, and A and C together paint B's house, it takes a grand total of 4 days for the sum of the three paint jobs. What's the smallest combined area of the three houses, in Planck lengths squared? Remember, the houses must have a positive integer area!

Exercise 10.3.1 - Do you think you could do a better job painting than these people?