Self-Descriptive Number Puzzles | 4 to 10 Digits | CAT Logical Reasoning

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Self-Descriptive Number Puzzles (4 to 10 Digits)

These are special numbers where each digit describes how many times a certain digit appears in the number itself.

Let’s explore them from 4-digit up to 10-digit numbers.

4-Digit Puzzle

Find a 4-digit number where:

Thousands place = number of 3s
Hundreds place = number of 2s
Tens place = number of 1s
Units place = number of 0s

Solution

Try 1210
Count:

1 zero → ✅ Matches 1st digit
2 ones → ✅ Matches 2nd digit
1 two → ✅ Matches 3rd digit
0 threes → ✅ Matches 4th digit

Try 2020
Count:

2 zeros → ✅ Matches 1st digit
0 ones → ✅ Matches 2nd digit
2 twos → ✅ Matches 3rd digit
0 threes → ✅ Matches 4th digit

Answer: 1210 and 2020

5-Digit Puzzle

Find a 5-digit number where:

Ten-thousands place = number of 0s
Thousands place = number of 1s
Hundreds place = number of 2s
Tens place = number of 3s
Units place = number of 4s

Solution

Try 21200
Count:

2 zeros → ✅
1 one → ✅
2 twos → ✅
0 threes → ✅
0 fours → ✅

Answer: 21200

6-Digit Puzzle

Find a 6-digit number where (from left):

Digit 1 (leftmost) = number of 0s
Digit 2 = number of 1s
Digit 3 = number of 2s
Digit 4 = number of 3s
Digit 5 = number of 4s
Digit 6 = number of 5s

Solution

No valid number exists for 6 digits in base-10 that satisfies the condition.
All possibilities fail logical verification.

Answer: No solution exists

7-Digit Puzzle

Find a 7-digit number where (from left):

Digit 1 (leftmost) = number of 0s
Digit 2 = number of 1s
Digit 3 = number of 2s
Digit 4 = number of 3s
Digit 5 = number of 4s
Digit 6 = number of 5s
Digit 7 = number of 6s

Solution

Try 3211000
Count:

3 zeros → ✅
2 ones → ✅
1 two → ✅
1 three → ✅
0 fours to sixes → ✅

Answer: 3211000

8-Digit Puzzle

Find an 8-digit number where (from left):

Digit 1 (leftmost) = number of 0s
Digit 2 = number of 1s
Digit 3 = number of 2s
Digit 4 = number of 3s
Digit 5 = number of 4s
Digit 6 = number of 5s
Digit 7 = number of 6s
Digit 8 = number of 7s

Solution

Try 42101000
Count:

4 zeros → ✅
2 ones → ✅
1 two → ✅
0 threes → ✅
1 four → ✅
0 fives to sevens → ✅

Answer: 42101000

9-Digit Puzzle

Find a 9-digit number where (from left):

Digit 1 = number of 0s
Digit 2 = number of 1s
Digit 3 = number of 2s
Digit 4 = number of 3s
Digit 5 = number of 4s
Digit 6 = number of 5s
Digit 7 = number of 6s
Digit 8 = number of 7s
Digit 9 = number of 8s

Solution

Try 521001000
Count:

5 zeros → ✅
2 ones → ✅
1 two → ✅
0 threes/fours → ✅
1 five → ✅
0 six to eight → ✅

Answer: 521001000

10-Digit Puzzle

Find a 10-digit number where (from left):

Digit 1 = number of 0s
Digit 2 = number of 1s
Digit 3 = number of 2s
Digit 4 = number of 3s
Digit 5 = number of 4s
Digit 6 = number of 5s
Digit 7 = number of 6s
Digit 8 = number of 7s
Digit 9 = number of 8s
Digit 10 = number of 9s

Solution

Try 6210001000
Count:

6 zeros → ✅
2 ones → ✅
1 two → ✅
0 threes to fives → ✅
1 six → ✅
0 seven to nine → ✅

Answer: 6210001000

🔧 Pattern Explanation (for Length ≥ 7)

For any base-b (i.e., b-digit self-descriptive number), when b ≥ 7, there is exactly one self-descriptive number. The pattern of digits looks like this:

(b − 4), 2, 1, 0, 0, ..., 0, 1, 0, 0, 0

The 1st digit is \(b - 4\) → count of 0s
The 2nd digit is 2 → count of 1s
The 3rd digit is 1 → count of 2s
The next few digits are 0
A 1 appears at the position \(b - 4\) → count of digit \(b - 4\)
All other digits are 0

Example (b = 9):

The number is: 5 2 1 0 0 1 0 0 0 → 521001000

🧠 Summary of Self-Descriptive Numbers

Digit Length	Valid Self-Descriptive Numbers
4	1210, 2020
5	21200
6	None
7	3211000
8	42101000
9	521001000
10	6210001000

💡 Final Insight

Self-descriptive numbers are not just quirky math trivia — they’re a brilliant intersection of logic, counting, and positional systems. For competitive exams like CAT, XAT, and aptitude interviews, questions like these test your lateral thinking and patience.

To Check More Sequence and Series Questions Click Here