Question

Is the statement is true if 24 is replaced by:

a) \(2\)

b) \(23\)

c) \(25\)

d) \(30\)

Short answer

(a) The resultant answer is True.

(b) The resultant answer is True.

(c) The resultant answer is True.

(d) The resultant answer is False.

Step-by-step solution

Given data

The given data is let \({x_i}\) denote the no. of matches the wrestler had played after the end of the \(i\)-th hour for \(1 \le i \le 75\), where \(1 \le {x_i} \le \)125. This also implies that \(1 + n \le {x_i} + n \le 125 + n\). Thus 150 positive integers \({x_1},{x_2}, \ldots ,{x_{75}},{x_1} + n,{x_2} + \) \(n, \ldots ,{x_{75}} + n\) are between 1 and \((125 + n)\), and only for \(n \le 24\) the Pigeonhole Principle implies that \(\exists i \ne j \ni {x_i} + \) \(n = {x_j}\), i.e. the wrestler played exactly \(n( \le 24)\) matches between the end of \(i\)-th and \(j\)-th hour.

Concept of Pigeonhole principle

If the number of pigeons exceeds the number of pigeonholes, at least one hole will hold at least two pigeons, according to the pigeon hole principle.

Simplify using pigeonhole principle

Observe that in case of \(n = 25\), all those 150 positive integers take values from between 1 and 150, so if there is no repetition then each is assigned value in the following order: \({x_1} = 1,{x_2} = 2, \ldots ,{x_{25}} = 25,{x_1} + 25 = 26,{x_2} + \)\(25 = 27, \ldots ,{x_{26}} = 51, \ldots \)

which implies even though PHP cannot be used here, exactly 25 matches have been played by the wrestler in first consecutive 25 hours.