How many pairs of digits in the number 613955 have as many numbers between…

Concept: For any two digits inside a number, the “numbers between them” can be counted two ways — by their position (how many digits sit between the pair inside the number) and by their value (how many natural numbers lie between the pair's digit-values in the 0–9 series). A pair qualifies only when these two counts are exactly equal. Since counting the terms between two values gives the same count whether the series is read forward (0→9) or backward (9→0), this equality is automatically direction-independent. By the established convention for this question type, a value-gap is counted only between two DISTINCT digit-values; two occurrences of the SAME digit are never treated as a qualifying pair, whatever their position gap, since there is no second distinct value to measure a natural-number gap against.

Application: Write 613955 with each digit's position — 6(1st), 1(2nd), 3(3rd), 9(4th), 5(5th), 5(6th) — and check every pair of distinct-value digits against both counts:

Cross-check: Re-measuring the same three matching pairs from the right-hand end of the number reproduces identical position gaps (2, 3 and 1), because a position gap between two fixed points does not change with the direction of counting — so the “forward and backward” requirement is satisfied by exactly these pairs and by no others. The one same-digit pair in the number — the two 5's — stays excluded on both readings by the same convention: there is no second distinct value to measure a gap against.

Result: Only three digit-pairs satisfy the condition, so three pairs qualify.