To prove that the bus number cannot be any smaller than 12 requires an exhaustive search.
No number strictly less than 12 can be written c_{1} + ... + c_{k} in two different
ways such that the products c_{1}c_{2}...c_{k} are equal. Also the only two
ways this can be done for 12 are 1,3,4,4 and 2,2,2,6 as given above.
This is easily checked by computer, but if anyone has a better argument, please let me know.