
Counting Problems
... Only the pairs (x1 , x3 ) = (2, 8), (5, 7), (8, 6), (11, 5), (14, 5), (17, 3), (20, 2), (23, 1), (26, 0). However, since 1 + x2 + x4 + x5 = x3 + x7 = x1 , it follows that x1 < 11. On the other hand, x1 > x3 . This means that only the pair (8, 6) can work. Indeed, x1 = 8, x3 = 6, x7 = 2, x6 = 2, and ...
... Only the pairs (x1 , x3 ) = (2, 8), (5, 7), (8, 6), (11, 5), (14, 5), (17, 3), (20, 2), (23, 1), (26, 0). However, since 1 + x2 + x4 + x5 = x3 + x7 = x1 , it follows that x1 < 11. On the other hand, x1 > x3 . This means that only the pair (8, 6) can work. Indeed, x1 = 8, x3 = 6, x7 = 2, x6 = 2, and ...