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

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.