If page $1$ is on the outside of the first sheet, then page $2$ is on the inside. Pages $3$ and $4$ are on the same sheet, as are pages $5$ and $6$. This therefore shares a sheet with page $7$, so is the inside of the sheet.
On the outside on the other side is therefore page $62$. The other three sheets have pages $63$ and $64$, $65$ and $66$ and $67$ and $68$ on them. Therefore there are $68$ pages, and four sheets per page, so $17$ sheets.
Alternatively, the sum of the page numbers on the same side of the sheet is always constant, since one side increases by $1$ every time the other decreases by $1$. This means the total is always $8 + 61 = 69$, so page $1$ shares with page $68$. Hence there are $68$ pages, so $17$ sheets.
This problem is taken from the UKMT Mathematical Challenges.