How to deduce this puzzle

Question

Every station on the railway system sells tickets to every other station. Some new stations were added. 46 sets of additional sets of tickets were required. How many new stations have been added? How many stations were there originally

Ans: 2 new stations (11 existing stations)

I don't know how to get to this answer. Please help me ?

Answer 1

Well, let's suppose that there used to be $n$ stations, and that $k$ new stations were added. Every station sells tickets to every other station, so prior to the new stations being added, each of the $n$ stations sold tickets to $n-1$ different destinations, meaning that there used to be $n(n-1)$ sets of tickets. After the new stations were added, there were $(n+k)(n+k-1)$ sets of tickets. Hence, the number of new sets of tickets is $$\begin{align}46 &= (n+k)(n+k-1)-n(n-1)\\ &= (n+k)^2-(n+k)-n^2+n\\ &= (n+k)^2-n^2+n-n-k\\ &= \bigl((n+k)+n\bigr)\bigl((n+k)-n\bigr)-k\\ &= (2n+k)k-k\\ &= (2n+k-1)k.\end{align}$$

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At this point the idea is to try to find a factor $k$ of $46$ such that $\frac{46}k=2n+k-1$ for some integer $n,$ or equivalently, such that $$\frac12\left(\frac{46}k-k+1\right)\tag{$\star$}$$ is an integer. Furthermore, since $n$ and $k$ are both numbers of train stations, it really doesn't make sense for $k$ or $n$ to be negative, and we certainly can't have $k=0$. So, we want to find a positive factor $k$ of $46$ such that $(\star)$ is a nonnegative integer. The positive factors of $46$ are $1,2,23,$ and $46,$ so there aren't too many cases to check. It turns out that $(\star)$ is an integer for all such values of $k,$ but it is negative for $k=23$ and $k=46.$ Consequently, we have $k=1$ or $k=2,$ which correspond (respectively) to $n=23$ and $n=11.$

Hence, one of the two following two must be true (hat tip to Gintas K for noticing that there were, in fact, two possibilities allowed by the chain of equations above):

• There were $23$ stations, and $1$ more station was added.
• There were $11$ stations, and $2$ more stations were added.

Now, if we take "some new stations were added" to mean that more than one station was added, then we can conclude that the second case must be the one that occurred.

Answer 2

When there are $n\geq1$ existing stations and $k\geq1$ new stations we need $2nk+k(k-1)$ new sets of tickets. From $$k(2n+k-1)=46$$ we conclude that $k\in\{1,2,23,46\}$. Trying $k=1$ we get $n=23$, trying $k=2$ we get $n=11$, and both cases are compatible with the story. On the other hand, the values $k=23$ and $k=46$ do not lead to an admissible $n$.