The fourteen digits of a credit card are to be written in the boxes shown…

Concept

If the sum of every three consecutive terms of a sequence always equals the same fixed constant, then moving the three-term window forward by one position must drop one term and pick up a new one without changing the total - so the term dropped must equal the term picked up. That means every term equals the term exactly three places after it, so the sequence of digits repeats after every three positions.

Application

1. Number the 14 boxes 1 to 14 in order. Group the positions by their remainder on dividing by 3: positions 1,4,7,10,13 form one group; positions 2,5,8,11,14 form a second group; positions 3,6,9,12 form a third group.

2. By the periodicity from the concept above, every box within the same group holds the same digit.

3. The box already filled with 7 belongs to the 1,4,7,10,13 group, so every box in that group holds 7. The box already filled with 8 belongs to the 3,6,9,12 group, so every box in that group holds 8. The box marked x belongs to the remaining 2,5,8,11,14 group, so every box in that group holds x.

4. Positions 6, 7 and 8 are three consecutive boxes, one from each group, so they hold 8, 7 and x respectively - and their sum must equal the fixed total of 18: 8 + 7 + x = 18, so x = 3.

Cross-check

Positions 3, 4 and 5 are a different three-consecutive window drawn from the same three groups (8, 7, x), and positions 12, 13 and 14 give a third such window - both independently confirm 8 + 7 + x = 18, so x = 3 consistently across the arrangement.

Hence x = 3.