Finding the 489th Digit in a Concatenated Number Sequence

The problem asks us to find the digit located at the 489th position in the infinite number sequence formed by concatenating all positive integers: 123456789101112... To solve this, we need to figure out which number the 489th digit belongs to and which digit it is within that number.

Analyzing Digit Count by Number Length

Let's count how many digits are used by numbers of different lengths:

• Single-digit numbers (1-9): There are 9 numbers (1, 2, ..., 9). Each uses 1 digit.
• Total digits from single-digit numbers = $9 \times 1 = 9$.
• These digits occupy positions 1 through 9 in the sequence.

• Two-digit numbers (10-99): There are $99 - 10 + 1 = 90$ numbers (10, 11, ..., 99). Each uses 2 digits.
• Total digits from two-digit numbers = $90 \times 2 = 180$.
• These digits occupy positions $9 + 1 = 10$ through $9 + 180 = 189$ in the sequence.

• Three-digit numbers (100-999): There are $999 - 100 + 1 = 900$ numbers (100, 101, ..., 999). Each uses 3 digits.
• Total digits from three-digit numbers = $900 \times 3 = 2700$.
• These digits occupy positions $189 + 1 = 190$ through $189 + 2700 = 2889$ in the sequence.

Locating the 489th Digit

We are looking for the 489th digit. Let's see where this position falls:

• After single-digit numbers, we've used 9 digits.
• After two-digit numbers, we've used $9 + 180 = 189$ digits.

Since 489 is greater than 189, the 489th digit must be part of a three-digit number.

Determining the Specific Three-Digit Number and Digit

The digits from three-digit numbers start after the 189th digit. We need to find the 489th digit overall, which is the $(489 - 189) = 300$th digit among the block of three-digit numbers.

Each three-digit number contributes 3 digits. To find which three-digit number the 300th digit belongs to, we divide 300 by 3:

Number of three-digit numbers covered by 300 digits = $\frac{300}{3} = 100$ numbers.

This means the 300th digit among the three-digit numbers is the very last digit contributed by the first 100 three-digit numbers concatenated together.

The sequence of three-digit numbers starts with 100. The first 100 three-digit numbers are:

$100, 101, 102, \dots, 100 + (100 - 1) = 199$.

The 100th three-digit number is 199.

The 489th digit is the 300th digit within the three-digit number sequence. This corresponds to the last digit of the 100th number in this sequence, which is 199.

The digits of 199 are 1, 9, and 9. The last digit (the 3rd digit) is 9.

Therefore, the 489th digit in the sequence 123456789101112... is 9.

Step-by-Step Solution

1. Count the total number of digits used by single-digit numbers (1-9). This is $9 \times 1 = 9$ digits. The 9th digit is 9.
2. Count the total number of digits used by two-digit numbers (10-99). This is $90 \times 2 = 180$ digits. The cumulative number of digits is $9 + 180 = 189$. The 189th digit is 9 (from 99).
3. Determine how many more digits are needed to reach the 489th position after the two-digit numbers. Remaining digits needed = $489 - 189 = 300$ digits.
4. Since we have passed position 189, the 489th digit is within the block of three-digit numbers.
5. Calculate how many three-digit numbers these 300 digits correspond to. Number of three-digit numbers = $\frac{300}{3} = 100$.
6. The 489th digit is the last digit of the 100th three-digit number in the sequence.
7. The 100th three-digit number (starting from 100) is $100 + (100 - 1) = 199$.
8. The 489th digit is the 3rd digit of the number 199, which is 9.

Revision Table - Understanding Digit Positions

Additional Information - General Approach to Finding Digits

This type of problem can be generalized to find the $N$th digit in the sequence of concatenated integers. The approach involves:

• Step 1: Calculate the total digits used by all single-digit numbers.
• Step 2: Calculate the total digits used by all two-digit numbers and add to the previous total.
• Step 3: Calculate the total digits used by all three-digit numbers and add to the previous total, and so on, until the cumulative total exceeds $N$.
• Step 4: Once you find the range of numbers (e.g., two-digit, three-digit) where the $N$th digit falls, subtract the cumulative digits from the previous ranges from $N$. This gives you the position of the digit within this block of numbers.
• Step 5: Divide the result from Step 4 by the number of digits per number in this block (e.g., 2 for two-digit numbers, 3 for three-digit numbers). The quotient tells you how many numbers are fully completed before the digit, and the remainder tells you which digit within the target number it is (remainder 1 means the 1st digit, 2 means the 2nd, 0 means the last digit of the previous number). Adjust calculation if the remainder is 0.
• Step 6: Identify the specific number and the specific digit within that number.