Perpetual calendar cubes keep track of the date all year around. They must be turned (or even transposed) once a day. The following is a spinoff problem I'm having trouble with. Any hints are much appreciated.
Label the sides of four cubes (rather than two as in the link above) with the digits $0,1,2,\cdots,9$ according to will. Just like the calendar cubes, turn them so that different integers are created. Determine the longest sequence of consecutive integers that can be created with 1, 2, 3 or 4 cubes (over all possible such labelings).
If there are several sequences of the same maximum length, find the "largest sequence" (i.e., a sequence in which the last term is $>$ any other last term in a longest sequence.).
I have nothing to show at this point. I can't even determine if my sequence of 33 integers using two cubes is a max length sequence!
Edit: Assume the digits are distinct so that $6$'s and $9$'s cannot be swapped.