Counting the Digit 3 from 1 to 1000: A Detailed Analysis

In this article, we will delve into the intriguing problem of counting how many times the digit 3 appears while writing the integers from 1 to 1000. We will break down the counting process by analyzing the occurrences of the digit 3 in each place value: the hundreds, tens, and units. We will also discuss an alternative approach using the J programming language and provide a concise Python program for those interested in implementation.

Counting the Digit 3 from 1 to 1000

The task is to determine the frequency of the digit 3 in the decimal representation of numbers from 1 to 1000. This can be achieved by systematically considering the numbers 0 to 999 and then addressing the number 1000 separately.

1. Counting 3s from 000 to 999

We will analyze the occurrences of the digit 3 in each digit place: hundreds, tens, and units.

Hundreds Place

The digit 3 can be in the hundreds place for the numbers 300 to 399.

From 300 to 399, there are 100 numbers. This gives us a total of 100 occurrences of the digit 3.

Tens Place

The digit 3 can appear in the tens place in the following ranges:

030-039 130-139 230-239 330-339 430-439 530-539 630-639 730-739 830-839 930-939

Each of these ranges has 10 occurrences of the digit 3. There are 10 such ranges.

This gives us a total of 100 occurrences of the digit 3.

Units Place

The digit 3 can be in the units place in the numbers from 003 to 993, occurring every set of 10 numbers (003, 013, 023, ..., 993).

Since there are 100 sets of 10 numbers, the number of 3s in the units place is 100.

Total from 000 to 999

Summing up the occurrences in each place:

Hundreds place: 100 occurrences Tens place: 100 occurrences Units place: 100 occurrences

Total occurrences of 3 from 000 to 999: 100 100 100 300

2. Counting 3s in 1000

The number 1000 does not contain the digit 3.

Final Total

The total occurrences of the digit 3 while writing the integers from 1 to 1000 is 300.

Brute Force Approach Using J Programming Language

Using the J programming language, a brute force approach can be implemented to verify the result:

/3sep 1 to 10000 4000

The answer is 4000 threes, but excluding 10000, the answer is 3000.

Another Approach: Using Python

Here is a single line Python program to find the number of times 3 appears from 1 to 1000:

print(sum(str(i).count('3') for i in range(1, 1001)))

This will output the total count of the digit 3.

Conclusion

Our analysis confirms that the digit 3 appears 300 times in the range from 1 to 1000. This methodical approach can be applied to similar problems in number theory, providing a solid foundation for understanding and solving similar digit frequency questions.