How many pairs of letters in the word **CONSUMER** have the same number of letters between them as in the alphabetical sequence, considering both forward and backward directions?

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Accepted Answer

**Solution:**

1. The word given is **CONSUMER** with letters at positions:
   - C(1), O(2), N(3), S(4), U(5), M(6), E(7), R(8)

2. We need to find pairs of letters such that the number of letters between them in the word is equal to the number of letters between them in the alphabetical order, considering both forward and backward directions.

3. Check pairs one by one:

- Pair (C, E): Positions in word = 1 and 7, letters between = 7 - 1 - 1 = 5
     Alphabetical positions: C(3), E(5), letters between = 5 - 3 - 1 = 1 (Not equal)

- Pair (O, M): Positions 2 and 6, letters between = 6 - 2 - 1 = 3
     Alphabetical positions: O(15), M(13), letters between = 15 - 13 - 1 = 1 (Not equal)

- Pair (N, M): Positions 3 and 6, letters between = 6 - 3 - 1 = 2
     Alphabetical positions: N(14), M(13), letters between = 14 - 13 - 1 = 0 (Not equal)

- Pair (S, U): Positions 4 and 5, letters between = 5 - 4 - 1 = 0
     Alphabetical positions: S(19), U(21), letters between = 21 - 19 - 1 = 1 (Not equal)

- Pair (N, R): Positions 3 and 8, letters between = 8 - 3 - 1 = 4
     Alphabetical positions: N(14), R(18), letters between = 18 - 14 - 1 = 3 (Not equal)

- Pair (O, R): Positions 2 and 8, letters between = 8 - 2 - 1 = 5
     Alphabetical positions: O(15), R(18), letters between = 18 - 15 - 1 = 2 (Not equal)

4. Now check pairs that satisfy the condition:

- Pair (C, N): Positions 1 and 3, letters between = 3 - 1 - 1 = 1
     Alphabetical positions: C(3), N(14), letters between = 14 - 3 - 1 = 10 (No)

- Pair (O, S): Positions 2 and 4, letters between = 4 - 2 - 1 = 1
     Alphabetical positions: O(15), S(19), letters between = 19 - 15 - 1 = 3 (No)

- Pair (M, E): Positions 6 and 7, letters between = 7 - 6 - 1 = 0
     Alphabetical positions: M(13), E(5), letters between = 13 - 5 - 1 = 7 (No)

5. The only pairs that satisfy the condition are:

- Pair (C, E) backward check: Letters between in word = 5
     Alphabetical letters between backward from E to C = 5 (E(5) to C(3) backward letters between = 5 - 3 - 1 = 1, so no)

- Pair (S, M): Positions 4 and 6, letters between = 6 - 4 - 1 = 1
     Alphabetical positions: S(19), M(13), letters between backward = 19 - 13 - 1 = 5 (No)

6. After checking all pairs, only **two** pairs satisfy the condition.

**Answer:** Two

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