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pigeonhole principle states that if n items are put into m containers, with n>m, then at least one container must contain more than one item.
Let ai , 1 ≤ i ≤ 77 be the number of games played till ith day, including the games played on the ith day. Then,
a1 < a2 < . . . < a77 ≤ 132 (1)
a1 + 21 < a2 + 21 < . . . < a77 + 21 ≤ 132 + 21 = 153 (2)
Note that there does not exist ai , aj such that i j, 1 ≤ i, j ≤ 77 and ai = aj . Similarly, in equation (2), a1 + 21 to a77 + 21 are distinct. It follows that there exist 2 × 77 = 154 summands (pigeons), and 153 distinct integer values (pigeonholes). Therefore, there exist two summands having the same value. i.e., ai = aj + 21 and this implies that ai − aj = 21. Thus, there exist a period of i−j consecutive days during which exactly 21 games are played, i.e., days j + 1, j + 2, . . . , i.