How many pairs of letters are there in the word **STRENGTHEN**, such that the number of letters between them in the word is equal to the number of letters between them in the English alphabet?

Question

Accepted Answer

**Solution:**

1. Write down the word with letter positions:

S(1) T(2) R(3) E(4) N(5) G(6) T(7) H(8) E(9) N(10)

2. We need to find pairs of letters where the number of letters between them in the word equals the number of letters between them in the English alphabet.

3. Calculate the difference in positions in the word and compare with the difference in alphabetical positions:

- Pair (S, R): Positions 1 and 3, letters between = 1 (position 2)
     Alphabet positions: S(19), R(18), difference = 1
     Matches.

- Pair (T, E): Positions 2 and 4, letters between = 1 (position 3)
     Alphabet positions: T(20), E(5), difference = 15
     Does not match.

- Pair (R, N): Positions 3 and 5, letters between = 1 (position 4)
     Alphabet positions: R(18), N(14), difference = 4
     Does not match.

- Pair (E, G): Positions 4 and 6, letters between = 1 (position 5)
     Alphabet positions: E(5), G(7), difference = 2
     Does not match.

- Pair (N, T): Positions 5 and 7, letters between = 1 (position 6)
     Alphabet positions: N(14), T(20), difference = 6
     Does not match.

- Pair (G, H): Positions 6 and 8, letters between = 1 (position 7)
     Alphabet positions: G(7), H(8), difference = 1
     Matches.

- Pair (T, E): Positions 7 and 9, letters between = 1 (position 8)
     Alphabet positions: T(20), E(5), difference = 15
     Does not match.

- Pair (H, N): Positions 8 and 10, letters between = 1 (position 9)
     Alphabet positions: H(8), N(14), difference = 6
     Does not match.

4. Check pairs with more letters between:

- Pair (S, E): Positions 1 and 4, letters between = 2 (positions 2,3)
     Alphabet positions: S(19), E(5), difference = 14
     No match.

- Pair (T, N): Positions 2 and 5, letters between = 2 (positions 3,4)
     Alphabet positions: T(20), N(14), difference = 6
     No match.

- Pair (R, G): Positions 3 and 6, letters between = 2 (positions 4,5)
     Alphabet positions: R(18), G(7), difference = 11
     No match.

- Pair (E, T): Positions 4 and 7, letters between = 2 (positions 5,6)
     Alphabet positions: E(5), T(20), difference = 15
     No match.

- Pair (N, H): Positions 5 and 8, letters between = 2 (positions 6,7)
     Alphabet positions: N(14), H(8), difference = 6
     No match.

- Pair (G, E): Positions 6 and 9, letters between = 2 (positions 7,8)
     Alphabet positions: G(7), E(5), difference = 2
     Matches.

- Pair (T, N): Positions 7 and 10, letters between = 2 (positions 8,9)
     Alphabet positions: T(20), N(14), difference = 6
     No match.

5. So, the pairs that satisfy the condition are:

- (S, R) with 1 letter between and 1 letter difference in alphabet
   - (G, H) with 1 letter between and 1 letter difference in alphabet
   - (G, E) with 2 letters between and 2 letters difference in alphabet

6. Total such pairs = 3

**Answer:** Three

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