# Salesman Pigeonhole Question

A salesman sells at least $$1$$ car each day for $$100$$ consecutive days selling a total of $$150$$ cars. Prove that for each value of $$n$$, $$1\le n \lt 50$$, there is a period of consecutive days during which he sold a total of exactly $$n$$ cars.

How do you prove this through Pigeonhole? Thanks in advance.

• What have you tried? Oct 2, 2019 at 3:43

Let $$T_k$$ be the total number of cars sold from days 1 to $$k$$.
The goal is to show that $$\{ T_k \} \cap \{ T_k + n \}$$ is non-empty.
The range of values is from 1 to $$150+n < 200$$. These are our holes. Hence, by PP, there is at one hole with at least 2 balls.
Can the 2 balls both come from $$\{T_k\}$$? (Explain why the answer is no.)
Can the 2 balls both come from $$\{T_k + n\}$$? (Explain why the answer is no.)
Hence, these 2 balls give us $$T_i = T_j + n$$.
So, from days $$j+1$$ to $$i$$, he sold $$n$$ cars.