How many such pairs of letters are there in the word “FOREIGN”, each of which…

Concept. A pair of letters qualifies when the number of letters lying between the two letters inside the word equals the number of letters lying between them in the English alphabet. If the letters occupy word positions i and j (with i less than j) and carry alphabet numbers a and b, then the word holds j − i − 1 letters between them and the alphabet holds |b − a| − 1, so the whole test collapses to one equality: j − i = |b − a|.

Direction. The stem asks only how many letters lie between the two letters, which is a size, not an order. A pair whose letters run downward through the alphabet therefore counts on exactly the same terms as one that runs upward, which is why the alphabet gap is taken in absolute value. Only an added phrase such as "in the same sequence as in the English alphabet" would narrow the scan to forward-running pairs.

Applying it to FOREIGN. First write each letter with its place in the word and its place in the alphabet.

Seven letters form 21 pairs. Sweep them by word gap, because the word gap is the smaller and more tightly bounded quantity, and for each pair compare it with the alphabet gap.

1. Word gap 1 (neighbouring letters): F–O gives an alphabet gap of 9, O–R gives 3, R–E gives 13, E–I gives 4, I–G gives 2, G–N gives 7. None of these equals 1.

2. Word gap 2: F–R gives 12, O–E gives 10, R–I gives 9, E–G gives 2, I–N gives 5. Here E–G matches, since 7 − 5 = 2 = 6 − 4.

3. Word gap 3: F–E gives 1, O–I gives 6, R–G gives 11, E–N gives 9. None of these equals 3.

4. Word gap 4: F–I gives 3, O–G gives 8, R–N gives 4. Here R–N matches, since |14 − 18| = 4 = 7 − 3.

5. Word gap 5: F–G gives 1 and O–N gives 1; word gap 6: F–N gives 8. None of these equals its target.

Cross-check by listing the in-between letters.

• E and G: the word places one letter, I, between them, and the alphabet places one letter, F, between E and G. One and one.

• R and N: the word places three letters, E, I and G, between them, and the alphabet places three letters, O, P and Q, between N and R. Three and three.

• All 21 pairs were swept above, so nothing qualifying was skipped, and the two matches found are the only ones.

Result. The word FOREIGN contains 2 such pairs of letters.