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Rashida wanted to invite her friends to her birthday and she was born in April. In a leap year, she started sending invitations from February 16 and completed them by March 15. At least one invitation was sent each day but no more than 50 invitations in total. Therefore, there has been a specific consecutive of days where exactly N invitations had to be sent. What is the value of N?

I started solving this problem in this way. As the year is a leap year, so February month has 29 days. So from February 16 to March 15, there are 28 days. At least one invitation was sent each day but no more than 50 invitations in total.

Therefore, there has been a specific consecutive of days where exactly N invitations had to be sent. I think the number is 8. Is this assumption correct?

Bill Dubuque
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Christopher Marlowe
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Using the pigeonhole principle, you can prove that there must be a string of consecutive days on which exactly $7$ invitations were issued.

Let $a_i$ be the cumulative number of invitations sent up to and including day $i$. Then $\{a_1, a_2, \ldots, a_{29}, a_1+7, a_2+7, \ldots, a_{29}+7 \}$ is a set of $58$ integers between $1$ and $57$, inclusive. Therefore, two of the integers, $a_i$ and $a_j+7$, must be the same. (Since at least one invitation was issued on each day, $i \neq j \Rightarrow a_i \neq a_j \Rightarrow a_i+7 \neq a_j+7$.) That means that exactly $7$ invitations must have been issued between days $j+1$ and $i$, inclusive.

Robert Shore
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