In the number ‘69573818’, how many pairs of the digits have the same number between them (both forward and backward direction) in the number series?

Correct Answer: More than four

To find the pairs of digits in the given number 69573818 that have the same number of digits between them as in the standard number series (in both forward and backward directions), let us analyze the digits and their positions step-by-step.

The given number is 69573818.

Let us check the pairs in both directions:

1. Forward Direction (Left to Right):

• Starting from digit 6:

- 6 to 7: Digits in sequence are 6, 9, 5, 7. Count from 6: 6 (position 1), 7 (position 2), 8 (position 3), 9 (position 4). Here, 7 is at position 4 (index gap 3), which doesn't match standard counting. Let us count normally: 6 → 7 (gap of 2 numbers: 9, 5). In the number series, between 6 and 7 there are 0 numbers. So, no match.

- 6 to 8: Counting from 6: 6, 7 (at 9), 8 (at 5)... wait, let us count forward numbers from 6:

- Position 1 (6): 6

- Position 2 (9): 7

- Position 3 (5): 8 -> Pair (5, 8)? Let us do systematic counting:

Let us count forward from each digit:

• From 6: 6, 7(9), 8(5), 9(7)... (No match)

• From 9: 9, 10, 11... (No digit higher than 9, so no match)

• From 5: 5, 6(7), 7(3), 8(8) → The digit 8 is at the 4th position after 5 in the number. In standard counting: 5, 6, 7, 8 (3 numbers between them). Here, between 5 and 8 (the first 8) are 7 and 3 (2 numbers). Wait, 5 (pos 3) and 8 (pos 6): distance is 6 - 3 = 3. Count: 5 -> 6(7), 7(3), 8(8). So (5, 8) is a pair!

• From 5: 5 (pos 3) to 8 (pos 8): distance is 8 - 3 = 5. Count: 5, 6(7), 7(3), 8(8), 9(1), 10(8) (No match)

• From 7: 7, 8(3), 9(8)... (No match)

• From 3: 3, 4(8), 5(1), 6(8)... (No match)

• From 8: 8, 9(1)... (No match)

• From 1: 1, 2(8)... (No match)

Now let us check pairs in detail for both directions by looking at actual number differences equal to positional distance:

Let positional indices be 1 to 8 for digits: 6(1), 9(2), 5(3), 7(4), 3(5), 8(6), 1(7), 8(8).

Let us evaluate positional difference equal to numerical difference $|d_j-d_i|=|j-i|$:

1. Pair (5, 7): Index of 5 is 3, index of 7 is 4. Numerical difference = 7 - 5 = 2. Positional difference = 4 - 3 = 1. (No)

2. Pair (5, 8): Index of 5 is 3, index of first 8 is 6. Numerical difference = 8 - 5 = 3. Positional difference = 6 - 3 = 3. Match! (Pair 1: 5 - 8)

3. Pair (7, 8): Index of 7 is 4, index of second 8 is 8. Numerical difference = 8 - 7 = 1. Positional difference = 8 - 4 = 4. (No)

4. Pair (3, 1): Backward direction: index of 3 is 5, index of 1 is 7. Numerical difference = 3 - 1 = 2. Positional difference = 7 - 5 = 2. Match! (Pair 2: 3 - 1)

5. Pair (7, 8): Index of 7 is 4, index of first 8 is 6. Numerical difference = 8 - 7 = 1. Positional difference = 6 - 4 = 2. (No)

6. Pair (9, 8): Index of 9 is 2, index of first 8 is 6. Difference = 9 - 8 = 1. Positional diff = 6 - 2 = 4. (No)

7. Pair (9, 8): Index of 9 is 2, index of second 8 is 8. Difference = 9 - 8 = 1. Positional diff = 8 - 2 = 6. (No)

8. Pair (5, 1): Index of 5 is 3, index of 1 is 7. Difference = 5 - 1 = 4. Positional diff = 7 - 3 = 4. Match! (Pair 3: 5 - 1)

9. Pair (6, 1): Index of 6 is 1, index of 1 is 7. Difference = 6 - 1 = 5. Positional diff = 7 - 1 = 6. (No)

10. Pair (7, 3): Index of 7 is 4, index of 3 is 5. Difference = 7 - 3 = 4. Positional diff = 5 - 4 = 1. (No)

11. Pair (9, 5): Backward: 9 (pos 2) and 5 (pos 3). Diff = 4, Pos diff = 1. (No)

12. Pair (8, 8): 8 (pos 6) and 8 (pos 8). Diff = 0, Pos diff = 2. (No)

13. Pair (6, 5): Index of 6 is 1, index of 5 is 3. Numerical difference = 6 - 5 = 1. Positional difference = 3 - 1 = 2. (No)

14. Pair (3, 8): Index of 3 is 5, index of second 8 is 8. Difference = 8 - 3 = 5. Positional difference = 8 - 5 = 3. (No)

15. Pair (7, 1): Index of 7 is 4, index of 1 is 7. Difference = 7 - 1 = 6. Positional difference = 7 - 4 = 3. (No)

16. Pair (6, 3): Index of 6 is 1, index of 3 is 5. Difference = 6 - 3 = 3. Positional difference = 5 - 1 = 4. (No)

17. Pair (9, 7): Index of 9 is 2, index of 7 is 4. Difference = 9 - 7 = 2. Positional diff = 4 - 2 = 2. Match! (Pair 4: 9 - 7)

18. Pair (6, 8): Index of 6 is 1, index of first 8 is 6. Difference = 8 - 6 = 2. Positional diff = 6 - 1 = 5. (No)

19. Pair (6, 8): Index of 6 is 1, index of second 8 is 8. Difference = 8 - 6 = 2. Positional diff = 8 - 1 = 7. (No)

20. Pair (9, 3): Index of 9 is 2, index of 3 is 5. Difference = 9 - 3 = 6. Positional diff = 5 - 2 = 3. (No)

21. Pair (8, 1): Index of first 8 is 6, index of 1 is 7. Difference = 8 - 1 = 7. Positional diff = 7 - 6 = 1. (No)

22. Pair (8, 1): Index of second 8 is 8, index of 1 is 7. Difference = 8 - 1 = 7. Positional diff = 8 - 7 = 1. (No)

23. Pair (5, 3): Index of 5 is 3, index of 3 is 5. Difference = 5 - 3 = 2. Positional diff = 5 - 3 = 2. Match! (Pair 5: 5 - 3)

Let us summary the matching pairs found:

1. (5, 8) - Forward: 5 (pos 3) to 8 (pos 6), diff = 3.

2. (3, 1) or (1, 3) - Backward: 3 (pos 5) to 1 (pos 7), diff = 2.

3. (5, 1) or (1, 5) - Backward: 5 (pos 3) to 1 (pos 7), diff = 4.

4. (9, 7) or (7, 9) - Backward: 9 (pos 2) to 7 (pos 4), diff = 2.

5. (5, 3) or (3, 5) - Backward: 5 (pos 3) to 3 (pos 5), diff = 2.

Since we have found 5 pairs, which is more than four, the correct answer option is More than four.