My school has many classes. Any two students share exactly one class. Any two classes share exactly one student. A class must have at minimum $3$ students, and there is at least one class with $17$ students. How many students attend my school?
How do I approach this question? How do I determine whether or not this question has an exact numerical answer?
This is difficult... help would be much appreciated.
You're seeking a projective plane with lines containing $17$ points (i.e., projective planes of order $16$). These have $16^2+16+1=273$ points (i.e., students) and $273$ lines (i.e., classes). There are a list $16$ inequivalent such projective planes listed here.
We need to check that every class has size $17$ and every student attends $17$ classes.
Let $C$ be the class of size $17$ and $x$ be a student not in $C$ (which exists, since there are "many" classes). From the assumptions, every class $x$ attends has a unique student in $C$ and any student $y \in C$ shares a unique class with $x$. Hence $x$ attends exactly $17$ classes.
Let $N$ be the set of classes attended by $x$ (we know $|N|=17$ from the above), and let $z$ be a class outside of $N$. From the assumptions, any student in $z$ shares a unique class in $N$ and any class in $N$ shares a unique student with $z$. Hence $z$ contains exactly $|N|=17$ students.
We now need to go back and repeat the argument for arbitrary $x$ and arbitrary $C$ for which $x \not\in C$ where we know $|C|=17$.
We conclude that every class has $17$ students and every student attends $17$ classes. Thus we indeed have a projective plane of order $16$.
The number of points (and lines) in a projective plane of order $n$ is $n^2+n+1$.