You are a landlord who owns a building that is an R x C grid of apartments; each apartment is a unit square cell with four walls. You want to rent out N
of these apartments to tenants, with exactly one tenant per apartment,
and leave the others empty. Unfortunately, all of your potential tenants
are noisy, so whenever any two occupied apartments share a wall (and
not just a corner), this will add one point of unhappiness to the
building. For example, a 2x2 building in which every apartment is
occupied has four walls that are shared by neighboring tenants, and so
the building's unhappiness score is 4.
If you place your N tenants optimally, what is the minimum unhappiness value for your building?
The first line of the input gives the number of test cases, T. T lines follow; each contains three space-separated integers: R, C, and N.
For each test case, output one line containing "Case #x: y", where x is
the test case number (starting from 1) and y is the minimum possible
unhappiness for the building.
1 ≤ T ≤ 1000.
0 ≤ N ≤ R*C.
1 ≤ R*C ≤ 16.
1 ≤ R*C ≤ 10000.
4 2 3 6 4 1 2 3 3 8 5 2 0
Case #1: 7 Case #2: 0 Case #3: 8 Case #4: 0