There are 15 digits in the first 12 counting numbers. That is, in the string of numbers:
1 2 3 4 5 6 7 8 9 10 11 12
there are 15 digits. What is the value of the last counting number if a string of
counting numbers, starting at 1, has 3425 digits?

To find the last counting number in a string of counting numbers that has 3425 digits, we can solve this problem by dividing it into cases based on the number of digits of the counting numbers:

### Step-by-Step Solution:

1. **Identify the structure of counting numbers by digit count:**
   - Numbers with **1 digit**: 1 to 9, which gives 9 numbers.
   - Numbers with **2 digits**: 10 to 99, which gives \(99 - 10 + 1 = 90\) numbers.
   - Numbers with **3 digits**: 100 to 999, which gives \(999 - 100 + 1 = 900\) numbers.
   - Numbers with **4 digits**: 1000 onward, until we reach 3425 digits.

2. **Calculate the total number of digits contributed by each group:**

   - **1-digit numbers**: There are 9 numbers, so they contribute:
     \[
     9 \times 1 = 9 \text{ digits}
     \]
     
   - **2-digit numbers**: There are 90 numbers, so they contribute:
     \[
     90 \times 2 = 180 \text{ digits}
     \]
     
   - **3-digit numbers**: There are 900 numbers, so they contribute:
     \[
     900 \times 3 = 2700 \text{ digits}
     \]

   - **4-digit numbers**: Starting from 1000, each has 4 digits. We use these numbers if the total count of digits has not yet reached 3425.

3. **Sum the digits contributed and determine when we reach 3425 digits:**

   - The digits from 1-digit and 2-digit numbers sum up to:
     \[
     9 + 180 = 189 \text{ digits}
     \]
   
   - Adding the 3-digit numbers, we get:
     \[
     189 + 2700 = 2889 \text{ digits}
     \]
   
   - This leaves:
     \[
     3425 - 2889 = 536 \text{ digits}
     \]
     remaining to reach 3425 digits.

4. **Determine how many 4-digit numbers we need:**

   - Each 4-digit number contributes 4 digits, so we need:
     \[
     \frac{536}{4} = 134 \text{ 4-digit numbers}
     \]
   
   - The first 4-digit number is 1000, so counting 134 numbers from there reaches:
     \[
     1000 + 134 - 1 = 1133
     \]

### Answer:
The last counting number in this sequence is **1133**.

Would you like a detailed explanation on any part, or do you have questions?

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Here are five related questions to deepen understanding:
1. How would this process change if we wanted to find the counting number for a different total number of digits?
2. How can we generalize this approach for any given digit target, not just 3425?
3. What would the solution look like if the digit count were limited to only 2-digit and 3-digit numbers?
4. How many total digits would there be if we included numbers up to 999?
5. What is the total number of counting numbers required to reach exactly 5000 digits?

**Tip:** When tackling problems involving large sequences, breaking them into cases by digit length or intervals often simplifies calculations.

There are 15 digits in the first 12 counting numbers. That is, in the string of numbers: 1 2 3 4 5 6 7 8 9 10 11 12 there are 15 digits. What is the value of the last counting number if a string of counting numbers, starting at 1, has 3425 digits?

Find the last counting number in a continuous sequence that has 3425 digits. Explore calculations involving counting numbers, digit contributions by number ranges, and sequences. Perfect for high school students learning number theory.

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