# Help an athlete impress a math geek

Help Lucy impress Nick the math geek and find his apartment number!

Lucy a track athlete, really likes Nick, a math geek. Nick is having a party at his apartment on Saturday and invites Lucy. Lucy knows where the apartment building is but does not know the apartment number

The invitation says:

Party starts 21:00 Saturday night. My 35th floor apartment is on the side of the floor with odd numbered apartments (the hallway on my floor is shaped like a large square with the odd numbered apartments on the outside of the outside and even numbered apartments on the inside).

Apartment numbers in my building start from 1 on ground one but do not necessarily start with the floor number (except floor 1) because all whole numbers (starting from 1) and used and there are not 100 apartments per floor.

The sum of the odd numbered apartments on my floor is 7975. Because I am so cool, I live in the apartment with the largest prime number on my floor.

Hint

Each floor has the same number of apartments

Hint 2

There are between 13 and 44 apartments on each floor

Hint 2

Don't ask Lucy to use her athleticism to run around the floor and knock on every odd number door to find him. Nick would consider that cheating

• I think 35th apartment should be 35th floor apartment. – Hugh Meyers Jun 10 '16 at 7:25
• Definitely. My first thought was "where are his other 34 apartments?". I only realised this means 35th floor when I read the answer that assumed this. Not sure if this is a localised colloquialism or a bad english problem but certainly in British English the meaning is not clear. – Chris Jun 10 '16 at 10:09
• Also, and utilized should be are used. – TRiG Jun 10 '16 at 10:28
• Idle observation: there's more space on the outside of a square than on the inside. Are the even-numbered apartments just smaller? – Ben Millwood Jun 10 '16 at 15:22
• All other elements aside, I appreciate a scenario posited in which the athlete is trying to win over the geek. – feelinferrety Jun 11 '16 at 1:23

$733$
You need a little more than $10$ numbers a little bigger than $700$.
How do we get that? Well, if you have n apartments per floor, you have about $n/2$ odd apartments which start at $34 * n + 1$. We have $13 < n > 44$. Take $n = 10$ as a lower bound because it is easy to do in your head. Five numbers between $340$ and $350$ gives a sum a little over $1,700$. Doubling $n$ gives you twice as many numbers and the numbers are twice as big, so $n = 20$ is going to be about $1,000$ too small. Increasing $n$ by one gives you one more odd apartment in the $700$ rage and bumps the others up by $34$. Plug $n = 21$ into a calculator and... surprise! It works! This gives the odd numbered apartments as $715$ to $735$. $735$ is obviously divisible by $5$, so $733$ is the largest prime.