Let $N$ be an odd square number. Show that $N$'s tens digit must be even.

How many numbers in this series are squares? $$1, 14, 144, 1444, 14444, \cdots$$

Find a square number which has two thousand and eighteen $6$s and some numbers of $0$s?

Find all pairs of integers $(x, y)$ such that $5\times (x^2 + 3)= y^2$.

There are $100$ lights lined up in a long room. Each light has its own switch and is currently off. The room has an entry door and an exit door. There are $100$ people lined up outside the entry door. Each light is numbered consecutively from $1$ to $100$. So is each person.

Person No. $1$ enters the room, switches on every light, and exits. Person No. $2$ enters and flips the switch on every second light (i.e. turn off lights $2$, $4$, $6$...). Person No. $3$ enters and flips the switch on every third light (i.e. toggle lights $3$, $6$, $9$...). This continues until all $100$ people have passed through the room. How many of the lights are on at the end?