To solve this problem, we need to identify the pattern in how the young girl is counting her fingers and use it to predict which finger she will end on when she reaches 2023.

Step 1: Understand the counting pattern.

• The girl counts in a "zigzag" manner, starting at the thumb and going to the little finger, then reversing back to the thumb.
• Let's assign numbers to the fingers as follows:
  • Thumb: 1
  • Index finger: 2
  • Middle finger: 3
  • Ring finger: 4
  • Little finger: 5

The pattern for the counting goes as follows:

1. Thumb (1)
2. Index finger (2)
3. Middle finger (3)
4. Ring finger (4)
5. Little finger (5)
6. Ring finger (6)
7. Middle finger (7)
8. Index finger (8)
9. Thumb (9)
10. Index finger (10)
11. Middle finger (11)
12. Ring finger (12)
13. Little finger (13)
14. Ring finger (14)
15. Middle finger (15)
16. Index finger (16)
17. Thumb (17)
18. Index finger (18)
19. Middle finger (19)
20. Ring finger (20)

The cycle is:

• Thumb → Index → Middle → Ring → Little → Ring → Middle → Index → Thumb → Index → Middle → Ring → Little → Ring → Middle → Index → Thumb, etc.

Step 2: Identify the cycle length.

From the pattern, you can see that the counting cycle repeats every 8 counts:

• 1st position (thumb) → 9th position (thumb) → 17th position (thumb), etc.

So, every 8 counts, the same fingers will be reached. This is a repeating cycle of 8.

Step 3: Determine the remainder when counting to 2023.

To find which finger corresponds to 2023, we need to determine where 2023 falls within this 8-count cycle. We do this by finding the remainder when 2023 is divided by 8:

$2023 \div 8 = 252 \text{ remainder } 7$

This means that after 252 full cycles of 8, we will be at position 7 in the next cycle.

Step 4: Identify which finger is in position 7.

From the pattern:

• Position 1: Thumb
• Position 2: Index
• Position 3: Middle
• Position 4: Ring
• Position 5: Little
• Position 6: Ring
• Position 7: Middle

Conclusion:

The mathematician predicted that her daughter would end on her middle finger when she counted to 2023. The remainder of 7 in the 8-cycle corresponds to the middle finger.

Would you like more details or have any questions?

Related questions:

1. What would be the finger for a count of 2040?
2. Can you derive a formula for determining the finger for any number in this zigzag pattern?
3. How does modular arithmetic help solve problems like this?
4. What is the longest cycle of a different finger-counting pattern you can think of?
5. How can this problem be generalized to patterns involving more than 5 fingers?

Tip:

Modular arithmetic is a powerful tool for solving cyclical problems, especially when a pattern repeats at regular intervals, like in this finger-counting puzzle.