How many such pairs of letters are there in the word **'TRANSFER'**, each of which has as many letters between them in the word as they have between them in the English alphabet?

Question

Accepted Answer

**Solution:**

1. Write down the word: **TRANSFER**

Positions: T(1), R(2), A(3), N(4), S(5), F(6), E(7), R(8)

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

3. Calculate the number of letters between each pair in the word:
   - For letters at positions *i* and *j*, number of letters between them in the word = |j - i| - 1

4. Calculate the number of letters between the same letters in the alphabet:
   - Alphabet positions: A=1, B=2, ..., Z=26
   - Number of letters between two letters = |AlphabetPosition(letter1) - AlphabetPosition(letter2)| - 1

5. Check all pairs:

- (T, R): positions 1 and 2
  - Letters between in word = 0
  - Alphabet positions: T=20, R=18
  - Letters between in alphabet = |20 - 18| - 1 = 1
  - Not equal

- (T, A): positions 1 and 3
  - Letters between in word = 1
  - Alphabet positions: T=20, A=1
  - Letters between in alphabet = |20 - 1| - 1 = 18
  - Not equal

- (T, N): positions 1 and 4
  - Letters between in word = 2
  - Alphabet positions: T=20, N=14
  - Letters between in alphabet = |20 - 14| - 1 = 5
  - Not equal

- (T, S): positions 1 and 5
  - Letters between in word = 3
  - Alphabet positions: T=20, S=19
  - Letters between in alphabet = |20 - 19| - 1 = 0
  - Not equal

- (T, F): positions 1 and 6
  - Letters between in word = 4
  - Alphabet positions: T=20, F=6
  - Letters between in alphabet = |20 - 6| - 1 = 13
  - Not equal

- (T, E): positions 1 and 7
  - Letters between in word = 5
  - Alphabet positions: T=20, E=5
  - Letters between in alphabet = |20 - 5| - 1 = 14
  - Not equal

- (T, R): positions 1 and 8
  - Letters between in word = 6
  - Alphabet positions: T=20, R=18
  - Letters between in alphabet = 1
  - Not equal

- (R, A): positions 2 and 3
  - Letters between in word = 0
  - Alphabet positions: R=18, A=1
  - Letters between in alphabet = 16
  - Not equal

- (R, N): positions 2 and 4
  - Letters between in word = 1
  - Alphabet positions: R=18, N=14
  - Letters between in alphabet = 3
  - Not equal

- (R, S): positions 2 and 5
  - Letters between in word = 2
  - Alphabet positions: R=18, S=19
  - Letters between in alphabet = 0
  - Not equal

- (R, F): positions 2 and 6
  - Letters between in word = 3
  - Alphabet positions: R=18, F=6
  - Letters between in alphabet = 11
  - Not equal

- (R, E): positions 2 and 7
  - Letters between in word = 4
  - Alphabet positions: R=18, E=5
  - Letters between in alphabet = 12
  - Not equal

- (R, R): positions 2 and 8
  - Letters between in word = 5
  - Alphabet positions: R=18, R=18
  - Letters between in alphabet = 0
  - Not equal

- (A, N): positions 3 and 4
  - Letters between in word = 0
  - Alphabet positions: A=1, N=14
  - Letters between in alphabet = 12
  - Not equal

- (A, S): positions 3 and 5
  - Letters between in word = 1
  - Alphabet positions: A=1, S=19
  - Letters between in alphabet = 17
  - Not equal

- (A, F): positions 3 and 6
  - Letters between in word = 2
  - Alphabet positions: A=1, F=6
  - Letters between in alphabet = 4
  - Not equal

- (A, E): positions 3 and 7
  - Letters between in word = 3
  - Alphabet positions: A=1, E=5
  - Letters between in alphabet = 3
  - Equal! (Pair 1)

- (A, R): positions 3 and 8
  - Letters between in word = 4
  - Alphabet positions: A=1, R=18
  - Letters between in alphabet = 16
  - Not equal

- (N, S): positions 4 and 5
  - Letters between in word = 0
  - Alphabet positions: N=14, S=19
  - Letters between in alphabet = 4
  - Not equal

- (N, F): positions 4 and 6
  - Letters between in word = 1
  - Alphabet positions: N=14, F=6
  - Letters between in alphabet = 7
  - Not equal

- (N, E): positions 4 and 7
  - Letters between in word = 2
  - Alphabet positions: N=14, E=5
  - Letters between in alphabet = 8
  - Not equal

- (N, R): positions 4 and 8
  - Letters between in word = 3
  - Alphabet positions: N=14, R=18
  - Letters between in alphabet = 3
  - Equal! (Pair 2)

- (S, F): positions 5 and 6
  - Letters between in word = 0
  - Alphabet positions: S=19, F=6
  - Letters between in alphabet = 12
  - Not equal

- (S, E): positions 5 and 7
  - Letters between in word = 1
  - Alphabet positions: S=19, E=5
  - Letters between in alphabet = 13
  - Not equal

- (S, R): positions 5 and 8
  - Letters between in word = 2
  - Alphabet positions: S=19, R=18
  - Letters between in alphabet = 0
  - Not equal

- (F, E): positions 6 and 7
  - Letters between in word = 0
  - Alphabet positions: F=6, E=5
  - Letters between in alphabet = 0
  - Equal! (Pair 3)

- (F, R): positions 6 and 8
  - Letters between in word = 1
  - Alphabet positions: F=6, R=18
  - Letters between in alphabet = 11
  - Not equal

- (E, R): positions 7 and 8
  - Letters between in word = 0
  - Alphabet positions: E=5, R=18
  - Letters between in alphabet = 12
  - Not equal

**Pairs found:**
- (A, E)
- (N, R)
- (F, E)

**Answer:** Three

Answer

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