A girl counts on the fingers of her left hand by calling the thumb 1, index finger 2, middle finger 3, ring finger 4, little finger 5, and then reversing direction (ring finger 6, middle finger 7, index finger 8, thumb 9, and so on); if she continues this pattern up to the number 1995, on which finger does she stop counting?

Introduction / Context:
This question is a classic arithmetic reasoning puzzle based on repeating patterns and modular arithmetic. The girl counts on her fingers forward and backward, creating a cyclical sequence of fingers that repeats after a fixed number of counts. We are asked to determine which finger corresponds to a large count, 1995, without listing all intermediate steps one by one. Recognizing the repeating cycle and using division with remainders is the key to efficient solving.

Given Data / Assumptions:
- The girl uses five fingers of her left hand: thumb, index, middle, ring, and little finger.
- She labels them in order as 1 thumb, 2 index, 3 middle, 4 ring, 5 little.
- After reaching the little finger, she reverses direction and continues 6 ring, 7 middle, 8 index, 9 thumb, then repeats the pattern.
- We assume she continues indefinitely in this forward backward manner until reaching 1995.

Concept / Approach:
The sequence of fingers follows a repeating pattern. Instead of tracking all 1995 counts, we identify the length of one full cycle and then compute the position of 1995 within that cycle using modular arithmetic. Once we know the remainder when 1995 is divided by the cycle length, we match that remainder to the corresponding finger in one cycle. This is a standard approach to problems involving periodic behaviour.

Step-by-Step Solution:
Step 1: Write down the sequence of fingers for the first few counts: 1 thumb, 2 index, 3 middle, 4 ring, 5 little, 6 ring, 7 middle, 8 index, 9 thumb, and then the pattern repeats. Step 2: Observe that the distinct finger sequence over one full cycle of counts is: thumb, index, middle, ring, little, ring, middle, index. This cycle has length 8. Step 3: Represent the cycle as positions 1 to 8: 1 thumb, 2 index, 3 middle, 4 ring, 5 little, 6 ring, 7 middle, 8 index. Step 4: To find the finger for count 1995, divide 1995 by 8, the cycle length, and find the remainder. Step 5: Perform the division: 1995 / 8 gives a quotient of 249 with a remainder of 3, since 8 * 249 = 1992 and 1995 - 1992 = 3. Step 6: A remainder of 3 means that the 1995th count falls on the same position in the cycle as count 3. Step 7: From the cycle, position 3 corresponds to the middle finger. Therefore the girl stops at the middle finger.

Verification / Alternative check:
We can verify the logic by checking a smaller example. For instance, count 11 also gives remainder 3 when divided by 8 (since 11 - 8 = 3), so the 11th count should also be on the middle finger. Listing quickly: 1 thumb, 2 index, 3 middle, 4 ring, 5 little, 6 ring, 7 middle, 8 index, 9 thumb, 10 index, 11 middle, which matches the pattern. This confirms that the remainder method correctly maps counts onto fingers, so applying it to 1995 is valid.

Why Other Options Are Wrong:
Option Thumb: The thumb corresponds to positions 1 and 9 in the extended pattern, which have remainder 1 when divided by 8, not remainder 3.
Option Index finger: The index finger appears at positions 2 and 8 in the cycle, associated with remainders 2 and 0 modulo 8, not remainder 3.
Option Ring finger: The ring finger appears at positions 4 and 6, corresponding to remainders 4 and 6. Neither matches 1995 modulo 8.
Option Little finger: The little finger appears only at position 5 in each cycle, which corresponds to remainder 5 and does not align with 1995 modulo 8.

Common Pitfalls:
A common error is to assume the cycle length is 5 instead of 8, ignoring the backward counting. Another mistake is miscalculating the remainder or mapping the remainder to the wrong position in the sequence. Some learners also attempt to simulate all counts, which is impractical and prone to mistakes. Recognizing periodicity and using modular arithmetic is the efficient and reliable approach for such problems.

Final Answer:
The girl ends counting on her middle finger.