How Letter Pairs Match Alphabet Gaps in "PRESTIGE"

This solution identifies pairs of letters within the word "PRESTIGE" that have the same number of intervening letters both within the word itself and in the standard English alphabet.

Step 1: Mapping Letters to Alphabetical Positions

First, we list the letters of the word "PRESTIGE" and their corresponding positions in the English alphabet (A=1, B=2, ..., Z=26).

Step 2: Finding Pairs with Matching Gaps (Forward Check)

We examine each letter and compare it with the subsequent letters in the word. The condition is that the count of letters between the pair in the word must equal the count of letters between them in the alphabet.

• Pair 1: (P, S)
  • P is at index 1 and S is at index 4 in "PRESTIGE".
  • Letters between P and S in the word: R, E (2 letters).
  • Alphabetical positions: P=16, S=19.
  • Letters between P and S in the alphabet: Q, R (2 letters). The calculation is $19 - 16 - 1 = 2$.
  • Since the number of letters matches (2 in the word vs. 2 in the alphabet), (P, S) is a valid pair.
• Pair 2: (P, T)
  • P is at index 1 and T is at index 5 in "PRESTIGE".
  • Letters between P and T in the word: R, E, S (3 letters).
  • Alphabetical positions: P=16, T=20.
  • Letters between P and T in the alphabet: Q, R, S (3 letters). The calculation is $20 - 16 - 1 = 3$.
  • Since the number of letters matches (3 in the word vs. 3 in the alphabet), (P, T) is a valid pair.
• Pair 3: (S, T)
  • S is at index 4 and T is at index 5 in "PRESTIGE".
  • Letters between S and T in the word: None (0 letters).
  • Alphabetical positions: S=19, T=20.
  • Letters between S and T in the alphabet: None (0 letters). The calculation is $20 - 19 - 1 = 0$.
  • Since the number of letters matches (0 in the word vs. 0 in the alphabet), (S, T) is a valid pair.

Step 3: Checking Other Potential Pairs

We continue checking other letter combinations to ensure accuracy. For instance:

• Consider the pair (E, G): E is at index 3, G is at index 7. There are 3 letters (S, T, I) between them in the word. Alphabetically, E (5) and G (7) have 1 letter (F) between them ($7 - 5 - 1 = 1$). Since $3 \neq 1$, this is not a valid pair.
• Consider the pair (I, G): I is at index 6, G is at index 7. There are 0 letters between them in the word. Alphabetically, G (7) and I (9) have 1 letter (H) between them ($9 - 7 - 1 = 1$). Since $0 \neq 1$, this is not a valid pair.

A thorough check of all forward and backward pairings confirms that only (P, S), (P, T), and (S, T) meet the specified criteria.

Step 4: Final Count

Therefore, there are 3 pairs of letters in the word "PRESTIGE" that satisfy the condition of having the same number of letters between them as in the alphabet.