Answer to Question #293955 in Discrete Mathematics for Tege

Question

A children watches TV at least one hours each days for seven weeks but never more than 11 hours in any week. prove that there is some period of consecutive days during in wich the children watches exactly 20 hours Of tv (if is assumed that the children watches Tv for a whole number of hours each days ?

Answer 1

Let a_i represent the number of hours spent by the child watching Tv within the first i days.

Taking 11 hours = one week, then for seven weeks is ( 11 * 7) = 77 hours'

Then 0 < a₁ < a₂ < < a₄₉ < 77

Since each of the seven weeks the child watches Tv at most 11 hours.

Also, all numbers in the sequence a₁ + 20, a₂ + 20, , a₄₉ + 20 are distinct and none of them exceeds 97.

But in the two sequences together, there are 98 numbers.

Thus it follows that from day ( j + 1) to day i, the child watches Tv for exactly 20 hours.