Let $a_1$ be the number of games played until day $1$, and so on, $a_i$ be the no games played until $i$.
Consider a sequence like $a_1,a_2, \dots a_{30}$ where $1≤ a_i ≤45, ∀a_i$.
Add $14$ to each elements of the sequence we get a new sequence $a_1+14,a_2+14, \dots a_{30}+14$ where $15≤ a_i+14 ≤59 , ∀a_i$.
Now we have two sequences
- $a_1,a_2, \dots, a_{30}$ and
- $a_1+14,a_2+14, \dots, a_{30}+14$
having $60$ elements in total with each elements taking a value $≤59$.
So according to pigeon hole principle there must be at least two elements taking a same value $≤59$ i.e., $a_i = a_j + 14$ for some $i$ and $j$.
There for there exists at least a period such as $a_j$ to $a_i$, in which $14$ matches are played.
