Probability with pigeon hole principle 
A gambler buys at least one lottery ticket everyday, maybe more.
   During the whole year he buys at most $400$ tickets. Show that during
   the year there exists a sequence of consecutive days for which the
   total number of tickets bought is exactly $330$. Hint: if $v_i$ is the
   number of tickets bought up to and including the $i^{th}$ day of the
   year, then we want to prove that there exists $j$ and $i$, with $j>i$,
   such that $v_j = v_{i-1} + 330$.

I'm not totally sure how to do this. However, I think that it involves the pigeon hole principle. thanks.
 A: The assertion is false.  
Counterexample: Let 
$$
v_i = 
\begin{cases}
2i - 1 & \text{if $1 \leq i \leq 18$}\\
i + 17 & \text{if $19 \leq i \leq 313$}\\
2i - 296 & \text{if $314 \leq i \leq 330$}\\
i + 35 & \text{if $331 \leq i \leq 365$}
\end{cases}
$$
Observe that if there were to exist $i, j$ such that $v_j = v_i + 330$, then $v_i \leq 35$.  In the counterexample, there are $18$ such numbers, each of which is odd.  Hence, for $1 \leq i \leq 18$, $v_i + 330$ is an odd number and $331 \leq v_i + 330 \leq 365$.  If $i \leq 313$, $v_i < 331$.  If $i \geq 331$, $v_i > 365$.  If $314 \leq i \leq 330$, $v_i$ is an even number satisfying $332 \leq v_i \leq 364$.  Hence, there is no $j$ such that $v_j = v_i + 330$.
Note:  If we were to replace the words "at most" with less or the year were a leap year, we could draw the desired conclusion.  What follows is an attempt to apply the Pigeonhole Principle with the given conditions.
Let $v_i$ be the total number of tickets the gambler has purchased after $i$ days.  Let $w_i = v_i + 330, 1 \leq i \leq 365$. Let
$$A = \{v_1, v_2, \ldots, v_{365}\}$$
Let 
$$B = \{w_1, w_2, \ldots, w_{365}\}$$
Since the gambler buys at least one lottery ticket each day, $|A| = |B| = 365$.  Since each $v_i$ satisfies the inequalities $1 \leq v_i \leq 400$, 
$$A \subseteq \{1, 2, 3, \ldots, 400\}$$ 
Since each $w_i$ satisfies the inequalities $331 \leq w_i \leq 730$, 
$$B \subseteq \{331, 332, 333, \ldots, 730\}$$
Hence, 
$$A \cup B \subseteq \{1, 2, 3, \ldots, 730\}$$
If $|A| + |B| \geq 731$, we could conclude that $A \cap B \neq \emptyset$, which would imply that there exists $v_j \in A$ and  $w_i \in B$ such that $v_j = w_i$, from which we could conclude that $v_j = v_i + 330$.  However, that is not the case here. 
