A girl counted in the following way on the fingers of her left hand:
She started by calling the thumb $1$, the index finger $2$, the middle finger $3$, the ring finger $4$, the little finger $5$ and then reversed direction calling the ring finger $6$, the middle finger $7$ and so on. She counted up to $1770$. She ended counting on which finger?

Let us look at the repeating pattern of counting on fingers:
1. Thumb
2. Index
3. Middle
4. Ring
5. Little
6. Ring
7. Middle
8. Index
9. Thumb
10. Index
...

Notice that the sequence completes a full forward-and-backward cycle every $8$ counts. The positions within the cycle of length $8$ correspond to the remainder when the count is divided by $8$:

* Remainder $1$: Thumb
* Remainder $2$ or $0$: Index finger
* Remainder $3$ or $7$: Middle finger
* Remainder $4$ or $6$: Ring finger
* Remainder $5$: Little finger

Let us find the remainder when $1770$ is divided by $8$:

$$1770=8\times 221+2$$

Since the remainder is $2$, it corresponds to the index finger.