5. On New Year’s day, 2023, a mathematician observed her young daugh- ter counting in a strange manner, using the five fingers of her left hand. She started by calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then she reversed direction, calling the ring finger 6, the middle fnger 7, the index finger 8, thumb 9, then back to the index finger for 10 middle finger 11, and so on.
She continued in this peculiar manner until she reached a count of 20 on her ring finger.
“What in the world are you doing?” her mother asked. The young girl stomped her foot.
“Now you’ve made me forget where I was. I’ll have to start all over agan. I’m counting up to 2023 to see what finger I’ll end on.”
The mathematician closed her eyes while she made a simple mental calculation. “You’ll end on your   ,” she said.
When the girl finished her count and found that her mother was right, she was so impressed by the predictive power of mathematics that she decided to work twice as hard on her arithmetic lessons.
How did the mother arrive at her prediction and what finger did she predict?

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.

On New Year’s day, 2023, a mathematician observed her young daughter counting in a strange manner, using the five fingers of her left hand. She started by calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then she reversed direction, calling the ring finger 6, the middle finger 7, the index finger 8, thumb 9, then back to the index finger for 10, middle finger 11, and so on. She continued in this peculiar manner until she reached a count of 20 on her ring finger. "What in the world are you doing?" her mother asked. The young girl stomped her foot. "Now you’ve made me forget where I was. I’ll have to start all over again. I’m counting up to 2023 to see what finger I’ll end on." The mathematician closed her eyes while she made a simple mental calculation. "You’ll end on your [finger]," she said. How did the mother arrive at her prediction and what finger did she predict?

Explore how modular arithmetic helps predict which finger a child will end on when counting up to 2023 in a cyclic pattern. This problem involves understanding cyclic sequences and using remainder calculations, suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation