EXAMPLE : During a month with 30 days, a baseball team plays at least one game a day, but no more
than 45 games. Show that there must be a period of some number of consecutive days during
which the team must play exactly 14 games.
(solution given in keneth rosen)
Solution: Let aj be the number of games played on or before the j th day of the month. Then
a1, a2, . . . , a30 is an increasing sequence of distinct positive integers, with 1 ≤ aj ≤ 45. Moreover,
a1 + 14, a2 + 14, . . . , a30 + 14 is also an increasing sequence of distinct positive integers,
with 15 ≤ aj + 14 ≤ 59.
The 60 positive integers a1, a2, . . . , a30, a1 + 14, a2 + 14, . . . , a30 + 14 are all less than
or equal to 59. Hence, by the pigeonhole principle two of these integers are equal. Because the
integers aj , j = 1, 2, . . . , 30 are all distinct and the integers aj + 14, j = 1, 2, . . . , 30 are all
distinct, there must be indices i and j with ai = aj + 14. This means that exactly 14 games
were played from day j + 1 to day i // IT SHOULD BE FROM j to i OR j+1 to day i is correct??
