How many pairs of letters are there in the word **TRANSFER**, such that the number of letters between them in the word is the same as the number of letters between them in the English alphabet?

Question

Accepted Answer

**Solution:**

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

2. Identify the position of each letter in the word:
   - T (1)
   - R (2)
   - A (3)
   - N (4)
   - S (5)
   - F (6)
   - E (7)
   - R (8)

3. For each pair of letters, calculate:
   - The number of letters between them in the word (difference in their positions minus 1)
   - The number of letters between them in the English alphabet (difference in alphabetical positions minus 1)

4. Alphabetical positions:
   - A = 1
   - E = 5
   - F = 6
   - N = 14
   - R = 18
   - S = 19
   - T = 20

5. Check pairs:

- (T, R): Positions 1 and 2, letters between in word = 0; Alphabet positions 20 and 18, letters between = 1 (20 - 18 - 1 = 1). Not equal.

- (T, A): Positions 1 and 3, letters between = 1; Alphabet positions 20 and 1, letters between = 18 (20 - 1 - 1 = 18). Not equal.

- (T, N): Positions 1 and 4, letters between = 2; Alphabet positions 20 and 14, letters between = 5 (20 - 14 - 1 = 5). Not equal.

- (T, S): Positions 1 and 5, letters between = 3; Alphabet positions 20 and 19, letters between = 0 (20 - 19 - 1 = 0). Not equal.

- (T, F): Positions 1 and 6, letters between = 4; Alphabet positions 20 and 6, letters between = 13 (20 - 6 - 1 = 13). Not equal.

- (T, E): Positions 1 and 7, letters between = 5; Alphabet positions 20 and 5, letters between = 14 (20 - 5 - 1 = 14). Not equal.

- (T, R): Positions 1 and 8, letters between = 6; Alphabet positions 20 and 18, letters between = 1 (20 - 18 - 1 = 1). Not equal.

- (R, A): Positions 2 and 3, letters between = 0; Alphabet positions 18 and 1, letters between = 16 (18 - 1 - 1 = 16). Not equal.

- (R, N): Positions 2 and 4, letters between = 1; Alphabet positions 18 and 14, letters between = 3 (18 - 14 - 1 = 3). Not equal.

- (R, S): Positions 2 and 5, letters between = 2; Alphabet positions 18 and 19, letters between = 0 (19 - 18 - 1 = 0). Not equal.

- (R, F): Positions 2 and 6, letters between = 3; Alphabet positions 18 and 6, letters between = 11 (18 - 6 - 1 = 11). Not equal.

- (R, E): Positions 2 and 7, letters between = 4; Alphabet positions 18 and 5, letters between = 12 (18 - 5 - 1 = 12). Not equal.

- (R, R): Positions 2 and 8, letters between = 5; Alphabet positions 18 and 18, letters between = 0. Not equal.

- (A, N): Positions 3 and 4, letters between = 0; Alphabet positions 1 and 14, letters between = 12 (14 - 1 - 1 = 12). Not equal.

- (A, S): Positions 3 and 5, letters between = 1; Alphabet positions 1 and 19, letters between = 17 (19 - 1 - 1 = 17). Not equal.

- (A, F): Positions 3 and 6, letters between = 2; Alphabet positions 1 and 6, letters between = 4 (6 - 1 - 1 = 4). Not equal.

- (A, E): Positions 3 and 7, letters between = 3; Alphabet positions 1 and 5, letters between = 3 (5 - 1 - 1 = 3). **Equal!**

- (A, R): Positions 3 and 8, letters between = 4; Alphabet positions 1 and 18, letters between = 16 (18 - 1 - 1 = 16). Not equal.

- (N, S): Positions 4 and 5, letters between = 0; Alphabet positions 14 and 19, letters between = 4 (19 - 14 - 1 = 4). Not equal.

- (N, F): Positions 4 and 6, letters between = 1; Alphabet positions 14 and 6, letters between = 7 (14 - 6 - 1 = 7). Not equal.

- (N, E): Positions 4 and 7, letters between = 2; Alphabet positions 14 and 5, letters between = 8 (14 - 5 - 1 = 8). Not equal.

- (N, R): Positions 4 and 8, letters between = 3; Alphabet positions 14 and 18, letters between = 3 (18 - 14 - 1 = 3). **Equal!**

- (S, F): Positions 5 and 6, letters between = 0; Alphabet positions 19 and 6, letters between = 12 (19 - 6 - 1 = 12). Not equal.

- (S, E): Positions 5 and 7, letters between = 1; Alphabet positions 19 and 5, letters between = 13 (19 - 5 - 1 = 13). Not equal.

- (S, R): Positions 5 and 8, letters between = 2; Alphabet positions 19 and 18, letters between = 0 (19 - 18 - 1 = 0). Not equal.

- (F, E): Positions 6 and 7, letters between = 0; Alphabet positions 6 and 5, letters between = 0 (6 - 5 - 1 = 0). **Equal!**

- (F, R): Positions 6 and 8, letters between = 1; Alphabet positions 6 and 18, letters between = 11 (18 - 6 - 1 = 11). Not equal.

- (E, R): Positions 7 and 8, letters between = 0; Alphabet positions 5 and 18, letters between = 12 (18 - 5 - 1 = 12). Not equal.

6. The pairs that satisfy the condition are:
   - (A, E)
   - (N, R)
   - (F, E)

**Answer:** There are **three** such pairs.

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