How many such pairs of digits are there in the number ‘95126139', which have as many digits between them in the number (both forward and backward direction) as they have between them in the Numeric Series?
Four
This question requires us to find pairs of digits within the number '95126139' that have a specific relationship compared to their spacing in the standard numeric sequence (1, 2, 3, 4, 5, 6, 7, 8, 9). We need to check this relationship in both forward and backward directions within the given number.
Understanding the Core Condition
The condition is that for a pair of digits (let's say $d_1$ and $d_2$), the number of digits physically located between them in the given number must be equal to the number of digits between them in the standard numeric series. The number of digits between $d_1$ and $d_2$ in the numeric series is calculated as $|d_1 - d_2| - 1$. We must find pairs where this count matches the count within the number '95126139'.
Analyzing the Number and Numeric Series
- The given number is: 95126139
- The numeric series digits are: 1, 2, 3, 4, 5, 6, 7, 8, 9.
- The number of digits between $d_1$ and $d_2$ in the numeric series is calculated using the formula: $\text{Count} = |d_1 - d_2| - 1$.
Identifying Pairs Matching the Condition
Let's systematically examine all possible unique pairs of digits present in '95126139' and check the condition. The digits present are 9, 5, 1, 2, 6, 1, 3, 9.
| Pair of Digits | Positions in '95126139' | Digits Between (In Number) | Digits Between (Numeric Series: $|d_1 - d_2| - 1$) | Condition Met? |
|---|---|---|---|---|
| (1, 2) | Forward: Pos 3 & 4 (12) Backward: Pos 4 & 3 (21) |
Forward: 0 Backward: 0 |
$|1 - 2| - 1 = 1 - 1 = 0$ | Yes |
| (1, 3) | Forward: Pos 3 & 7 (1..3); Pos 6 & 7 (13) Backward: Pos 7 & 3 (3..1); Pos 7 & 6 (31) |
Forward (3,7): 3 digits (2, 6, 1) Forward (6,7): 0 digits Backward (7,3): 3 digits (1, 6, 2) Backward (7,6): 0 digits |
$|1 - 3| - 1 = 2 - 1 = 1$ | No |
| (1, 5) | Forward: Pos 3 & 2 (No); Pos 6 & 2 (No) Backward: Pos 3 & 2 (15); Pos 6 & 2 (1..5) |
Backward (3,2): 0 digits Backward (6,2): 3 digits (6, 2, 1) -- Error in manual check, positions are 6 and 2. Number is 9 5 1 2 6 1 3 9. Backward from 6 to 2: 6, 1, 2. Digits between: 1, 2. Count = 2. Let's recheck calculation: Pos 2 (5), Pos 6 (1). Between = 1, 2, 6. Count = 3. Correct. |
$|1 - 5| - 1 = 4 - 1 = 3$ | Yes (Pair: 5, 1 at positions 2, 6) |
| (1, 6) | Forward: Pos 3 & 5 (126); Pos 6 & 5 (No) Backward: Pos 5 & 3 (6..1); Pos 5 & 6 (61) |
Forward (3,5): 1 digit (2) Backward (5,3): 1 digit (2) |
$|1 - 6| - 1 = 5 - 1 = 4$ | No |
| (1, 9) | Forward: Pos 3 & 1 (No); Pos 3 & 8 (1..9); Pos 6 & 1 (No); Pos 6 & 8 (19) Backward: Pos 1 & 3 (9..1); Pos 8 & 3 (9..1); Pos 1 & 6 (9..1); Pos 8 & 6 (9..1) |
Forward (3,8): 4 digits (2, 6, 1, 3) Forward (6,8): 1 digit (3) Backward (1,3): 1 digit (5) Backward (8,3): 4 digits (1, 6, 1, 3) Backward (1,6): 4 digits (5, 1, 2, 6) Backward (8,6): 2 digits (1, 3) |
$|1 - 9| - 1 = 8 - 1 = 7$ | No |
| (2, 3) | Forward: Pos 4 & 7 (2613) Backward: Pos 7 & 4 (3162) |
Forward: 2 digits (6, 1) Backward: 2 digits (1, 6) |
$|2 - 3| - 1 = 1 - 1 = 0$ | No |
| (2, 5) | Forward: Pos 4 & 2 (No) Backward: Pos 4 & 2 (215) |
Backward: 1 digit (1) | $|2 - 5| - 1 = 3 - 1 = 2$ | No |
| (2, 6) | Forward: Pos 4 & 5 (26) Backward: Pos 5 & 4 (62) |
Forward: 0 digits Backward: 0 digits |
$|2 - 6| - 1 = 4 - 1 = 3$ | No |
| (2, 9) | Forward: Pos 4 & 1 (No); Pos 4 & 8 (2..9) Backward: Pos 1 & 4 (9..2); Pos 8 & 4 (9..2) |
Forward (4,8): 3 digits (6, 1, 3) Backward (1,4): 2 digits (5, 1) Backward (8,4): 3 digits (1, 3, 1, 6) -- Error in manual check. Backward positions 8 and 4. Digits between: 3, 1, 6. Count = 3. Correct. |
$|2 - 9| - 1 = 7 - 1 = 6$ | No |
| (3, 5) | Forward: Pos 7 & 2 (No) Backward: Pos 7 & 2 (3..5) |
Backward: 4 digits (1, 6, 2, 1) | $|3 - 5| - 1 = 2 - 1 = 1$ | No |
| (3, 6) | Forward: Pos 7 & 5 (No) Backward: Pos 7 & 5 (316) |
Backward: 1 digit (1) | $|3 - 6| - 1 = 3 - 1 = 2$ | No |
| (3, 9) | Forward: Pos 7 & 1 (No); Pos 7 & 8 (39) Backward: Pos 1 & 7 (9..3); Pos 7 & 1 (3..9) |
Forward (7,1): 5 digits (5, 1, 2, 6, 1) Forward (7,8): 0 digits Backward (1,7): 5 digits (5, 1, 2, 6, 1) Backward (7,1): 5 digits (1, 6, 2, 1, 5) |
$|3 - 9| - 1 = 6 - 1 = 5$ | Yes (Pair: 9, 3 at positions 1, 7) |
| (5, 6) | Forward: Pos 2 & 5 (5126) Backward: Pos 5 & 2 (6215) |
Forward: 2 digits (1, 2) Backward: 2 digits (2, 1) |
$|5 - 6| - 1 = 1 - 1 = 0$ | No |
| (5, 9) | Forward: Pos 2 & 1 (No); Pos 2 & 8 (5..9) Backward: Pos 1 & 2 (95); Pos 8 & 2 (9..5) |
Forward (2,8): 5 digits (1, 2, 6, 1, 3) Backward (1,2): 0 digits Backward (8,2): 5 digits (1, 2, 6, 1, 3) -- error in manual check. Backward pos 8 and 2. Digits between: 3, 1, 6, 2, 1. Count = 5. Correct. |
$|5 - 9| - 1 = 4 - 1 = 3$ | No |
| (6, 9) | Forward: Pos 5 & 1 (No); Pos 5 & 8 (6139) Backward: Pos 1 & 5 (9..6); Pos 8 & 5 (9..6) |
Forward (5,8): 2 digits (1, 3) Backward (1,5): 3 digits (5, 1, 2) Backward (8,5): 2 digits (3, 1) |
$|6 - 9| - 1 = 3 - 1 = 2$ | Yes (Pair: 6, 9 at positions 5, 8) |
The pairs that satisfy the condition are:
- Pair (1, 2): Found at positions 3 and 4 (12). 0 digits between them in the number. In the numeric series, $|1 - 2| - 1 = 0$.
- Pair (5, 1): Found at positions 2 and 6 (51261). 3 digits (1, 2, 6) are between them in the number. In the numeric series, $|5 - 1| - 1 = 3$.
- Pair (9, 3): Found at positions 1 and 7 (9512613). 5 digits (5, 1, 2, 6, 1) are between them in the number. In the numeric series, $|9 - 3| - 1 = 5$.
- Pair (6, 9): Found at positions 5 and 8 (6139). 2 digits (1, 3) are between them in the number. In the numeric series, $|6 - 9| - 1 = 2$.
These checks consider both forward and backward directions implicitly by looking at all unique pairs. For example, checking (1, 2) forward also covers checking (2, 1) backward.
Conclusion
By analyzing all possible pairs of digits in the number '95126139' against the numeric series rule, we found 4 pairs that satisfy the given condition. These pairs are (1, 2), (1, 5), (3, 9), and (6, 9).
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