X X X 7 X X X y X X X 8 X X The fourteen digits of a credit card are to be…

Concept: If the sum of every three CONSECUTIVE terms of a sequence is the same constant, then subtracting one three-term window from the next (overlapping) window shows that terms three positions apart must be equal - the sequence repeats with period 3.

1. Number the fourteen digits from position 1 to position 14. The digit 7 is at position 4, the digit y is at position 8, and the digit 8 is at position 12.

2. Since every three consecutive digits sum to 18, digits three positions apart are equal, so the fourteen positions split into three repeating groups: positions 1, 4, 7, 10, 13 form one group; positions 2, 5, 8, 11, 14 form a second group; and positions 3, 6, 9, 12 form a third group.

3. Position 4 (value 7) belongs to the first group, so every position in that group also equals 7.

4. Position 12 (value 8) belongs to the third group, so every position in that group also equals 8.

5. Position 8 (value y) belongs to the second group, so every position in that group also equals y.

6. Any three consecutive positions contain exactly one position from each of the three groups, so their sum is always 7 + y + 8.

7. Since this sum must equal 18: 7 + y + 8 = 18, which gives y = 3.

Cross-check: positions 12, 13 and 14 belong to the third, first and second groups respectively, giving 8 + 7 + 3 = 18 - the same total holds for every block of three consecutive digits across all fourteen digits, confirming the value.