Find the unique 10-digit self-descriptive number in which the 1st digit counts zeros, the 2nd counts ones, … and the 10th digit counts nines in the number itself.

Introduction / Context:
This puzzle asks you to find a very special 10-digit number with a self-descriptive property. Each position in the number tells you how many times a particular digit appears in the entire number. The first digit tells you how many zeros there are, the second digit tells you how many ones there are, the third digit tells you how many twos there are, and so on, until the tenth digit which tells you how many nines there are. Numbers like this are called self descriptive numbers and they are an excellent test of logical reasoning, counting, and consistency checking.

Given Data / Assumptions:

• The number must be exactly 10 digits long.
• The 1st digit equals the total count of digit 0 in the entire 10-digit number.
• The 2nd digit equals the total count of digit 1 in the number.
• The 3rd digit equals the total count of digit 2, and so on.
• The 10th digit equals the total count of digit 9 in the number.
• All counts must be non negative integers, and the total of all counts must be 10, because the number has 10 digits in total.

Concept / Approach:
The concept is to build a self consistent system of counts. We are really solving a system of conditions: the digit in position i tells us how many of digit (i - 1) appear. For 10 digits, the counts of 0 through 9 must add up to 10. In known self descriptive numbers in base 10, most digits are zero, and only a few are non zero counts. A standard solution is 6210001000. In this candidate, the digits from left to right are 6, 2, 1, 0, 0, 0, 1, 0, 0, 0. We must verify that this 10-digit value satisfies the rules that define the puzzle.

Step-by-Step Solution:
Step 1: Write the candidate number clearly: 6210001000. Step 2: Count how many zeros appear in the entire number. The digits that are not zero are 6, 2, 1, and 1 (four digits). The total length is 10, so there are 10 - 4 = 6 zeros. The first digit is 6, so the count of zeros matches the first digit. Step 3: Count how many ones appear. Looking through 6210001000, there are exactly two digits 1. The second digit of the number is 2, which correctly records that there are two ones. Step 4: Count how many twos appear. There is exactly one digit 2 in the whole number. The third digit is 1, which correctly records one two. Step 5: Count how many threes appear. There are no 3 digits in 6210001000. The fourth digit is 0, so the count of threes is correctly recorded as zero. Step 6: Similarly, count how many fours appear. There are no 4 digits, and the fifth digit is 0, so this also matches. Step 7: Count how many fives appear. Again, there are no 5 digits, and the sixth digit is 0, which is consistent. Step 8: Count how many sixes appear. There is exactly one 6 in the entire number, at the first position. The seventh digit is 1, so the count of sixes is correctly recorded. Step 9: Count how many sevens, eights, and nines appear. None of these digits occur anywhere in 6210001000, so each of their counts is 0. The eighth, ninth, and tenth digits are 0, 0, and 0, which is fully consistent. Step 10: Check that the sum of all the counts equals 10: 6 (zeros) + 2 (ones) + 1 (twos) + 0 + 0 + 0 + 1 (sixes) + 0 + 0 + 0 = 10. This matches the total number of digits, further confirming that the number is valid.
Verification / Alternative check:
To verify more systematically, you can write a small table with digits 0 to 9 in one row and the corresponding counts implied by 6210001000 in the second row. For each digit, check the actual frequency in the number. You will find that digit 0 occurs six times, digit 1 occurs two times, digit 2 occurs once, digits 3, 4, and 5 occur zero times, digit 6 occurs once, and digits 7, 8, and 9 occur zero times. These observed counts exactly match the digits 6, 2, 1, 0, 0, 0, 1, 0, 0, 0 in the candidate number. Because all conditions are satisfied, 6210001000 is a self descriptive number and answers the puzzle completely.

Why Other Options Are Wrong:
In 5210011000, the first digit claims that there are five zeros, but a quick count reveals a different number of zeros, so it fails the very first condition. In 4210102000 and 6110002000, similar inconsistencies appear: for example the claimed numbers of ones or twos do not match their actual frequencies in the number. Any deviation between a claimed count and the real number of occurrences immediately disqualifies a candidate. Only 6210001000 survives all such checks.

Common Pitfalls:
A common mistake is to try to guess a pattern rather than check counts carefully. Many learners also forget that the total of all digits in the number must equal 10, because they are counting how often each digit from 0 to 9 appears. Another pitfall is to miscount zeros, especially when they appear in long blocks. A reliable method is to list all digits, circle non zero entries, and then calculate the number of zeros as the remainder. Being systematic and patient is essential in self descriptive number puzzles.

Final Answer:
The required 10-digit self descriptive number is 6210001000, where each position correctly counts how many times its corresponding digit appears in the entire number.