0  0  0  0  0
 1  0  0  0  0
 1  2  0  0  0
 1  2  3  0  0
 1  2  3  4  0
 1  2  3  4  5  # first possibly valid set, though having
 1  2  3  4  6    # 1 2 3 be from the same school lets you
 1  2  3  4  7    # quickly skip obviously invalid matchings
 ...              # getting to something like
 1  4  7  11 14   # <- fairly quickly
 ...              
 1  2  3  4 105   
 ...
 1  2  3 104 105
 ...
101 102 103 104 105 # last possible value