Solving the Word Puzzle: Letter Interchange and Pair Counting

The problem asks us to first modify the word "SPIRITUAL" by interchanging certain letters and then count the pairs of letters in the resulting new word that have the same number of letters between them as they do in the standard English alphabet series.

Step-by-Step Letter Interchange

Let's start with the original word and its letter positions:

Original Word: S P I R I T U A L

Positions:       1 2 3 4 5 6 7 8 9

The first instruction is to interchange the position of the second and fifth letters.

• The second letter is 'P'.
• The fifth letter is 'I'.

Interchanging these gives us:

Word after 1st Interchange: S I I R P T U A L

Positions:                     1 2 3 4 5 6 7 8 9

The second instruction is to interchange the position of the fourth and sixth letters with the seventh and ninth letters respectively.

• The fourth letter is 'R', which is interchanged with the seventh letter 'U'.
• The sixth letter is 'T', which is interchanged with the ninth letter 'L'.

Applying these interchanges to the word from the previous step:

Word before 2nd Interchange: S I I R P T U A L

Letters involved:                  ^     ^  ^  ^

Positions involved:                4     6  7  9

Applying the swaps (R <--> U, T <--> L):

New Word: S I I U P L R A T

Positions:      1 2 3 4 5 6 7 8 9

The new word after all interchanges is "SIIUPLRAT".

Counting Letter Pairs in the New Word

Now, we need to find pairs of letters in the word "SIIUPLRAT" that satisfy the condition: the number of letters between them in the word is equal to the number of letters between them in the English alphabet.

Let's write down the new word and the alphabetical position of each letter:

We will check for pairs in both forward and backward directions. The number of letters between two letters at positions \(p_1\) and \(p_2\) in the word is \(|p_1 - p_2| - 1\). The number of letters between two letters with alphabetical values \(v_1\) and \(v_2\) in the alphabet is \(|v_1 - v_2| - 1\).

Let's check each pair:

• S (19) and I (9) at positions 1 and 2: Word gap = \(|1-2| - 1 = 0\). Alphabet gap = \(|19-9| - 1 = 10 - 1 = 9\). No match.
• S (19) and I (9) at positions 1 and 3: Word gap = \(|1-3| - 1 = 1\). Alphabet gap = \(|19-9| - 1 = 9\). No match.
• S (19) and U (21) at positions 1 and 4: Word gap = \(|1-4| - 1 = 2\). Alphabet gap = \(|19-21| - 1 = 2 - 1 = 1\). No match.
• S (19) and P (16) at positions 1 and 5: Word gap = \(|1-5| - 1 = 3\). Alphabet gap = \(|19-16| - 1 = 3 - 1 = 2\). No match.
• S (19) and L (12) at positions 1 and 6: Word gap = \(|1-6| - 1 = 4\). Alphabet gap = \(|19-12| - 1 = 7 - 1 = 6\). No match.
• S (19) and R (18) at positions 1 and 7: Word gap = \(|1-7| - 1 = 5\). Alphabet gap = \(|19-18| - 1 = 1 - 1 = 0\). No match.
• S (19) and A (1) at positions 1 and 8: Word gap = \(|1-8| - 1 = 6\). Alphabet gap = \(|19-1| - 1 = 18 - 1 = 17\). No match.
• S (19) and T (20) at positions 1 and 9: Word gap = \(|1-9| - 1 = 7\). Alphabet gap = \(|19-20| - 1 = 1 - 1 = 0\). No match.

Checking from the second letter (I at pos 2):

• I (9) at pos 2 and I (9) at pos 3: Word gap = \(|2-3| - 1 = 0\). Alphabet gap = \(|9-9| - 1 = -1\). No match (cannot have negative letters).
• I (9) at pos 2 and U (21) at pos 4: Word gap = \(|2-4| - 1 = 1\). Alphabet gap = \(|9-21| - 1 = 12 - 1 = 11\). No match.
• I (9) at pos 2 and P (16) at pos 5: Word gap = \(|2-5| - 1 = 2\). Alphabet gap = \(|9-16| - 1 = 7 - 1 = 6\). No match.
• I (9) at pos 2 and L (12) at pos 6: Word gap = \(|2-6| - 1 = 3\). Alphabet gap = \(|9-12| - 1 = 3 - 1 = 2\). No match.
• I (9) at pos 2 and R (18) at pos 7: Word gap = \(|2-7| - 1 = 4\). Alphabet gap = \(|9-18| - 1 = 9 - 1 = 8\). No match.
• I (9) at pos 2 and A (1) at pos 8: Word gap = \(|2-8| - 1 = 5\). Alphabet gap = \(|9-1| - 1 = 8 - 1 = 7\). No match.
• I (9) at pos 2 and T (20) at pos 9: Word gap = \(|2-9| - 1 = 6\). Alphabet gap = \(|9-20| - 1 = 11 - 1 = 10\). No match.

Checking from the third letter (I at pos 3):

• I (9) at pos 3 and U (21) at pos 4: Word gap = \(|3-4| - 1 = 0\). Alphabet gap = \(|9-21| - 1 = 11\). No match.
• I (9) at pos 3 and P (16) at pos 5: Word gap = \(|3-5| - 1 = 1\). Alphabet gap = \(|9-16| - 1 = 6\). No match.
• I (9) at pos 3 and L (12) at pos 6: Word gap = \(|3-6| - 1 = 2\). Alphabet gap = \(|9-12| - 1 = 2\). Match! (I-L)
• I (9) at pos 3 and R (18) at pos 7: Word gap = \(|3-7| - 1 = 3\). Alphabet gap = \(|9-18| - 1 = 8\). No match.
• I (9) at pos 3 and A (1) at pos 8: Word gap = \(|3-8| - 1 = 4\). Alphabet gap = \(|9-1| - 1 = 7\). No match.
• I (9) at pos 3 and T (20) at pos 9: Word gap = \(|3-9| - 1 = 5\). Alphabet gap = \(|9-20| - 1 = 10\). No match.

Checking from the fourth letter (U at pos 4):

• U (21) at pos 4 and P (16) at pos 5: Word gap = \(|4-5| - 1 = 0\). Alphabet gap = \(|21-16| - 1 = 4\). No match.
• U (21) at pos 4 and L (12) at pos 6: Word gap = \(|4-6| - 1 = 1\). Alphabet gap = \(|21-12| - 1 = 8\). No match.
• U (21) at pos 4 and R (18) at pos 7: Word gap = \(|4-7| - 1 = 2\). Alphabet gap = \(|21-18| - 1 = 2\). Match! (U-R)
• U (21) at pos 4 and A (1) at pos 8: Word gap = \(|4-8| - 1 = 3\). Alphabet gap = \(|21-1| - 1 = 19\). No match.
• U (21) at pos 4 and T (20) at pos 9: Word gap = \(|4-9| - 1 = 4\). Alphabet gap = \(|21-20| - 1 = 0\). No match.

Checking from the fifth letter (P at pos 5):

• P (16) at pos 5 and L (12) at pos 6: Word gap = \(|5-6| - 1 = 0\). Alphabet gap = \(|16-12| - 1 = 3\). No match.
• P (16) at pos 5 and R (18) at pos 7: Word gap = \(|5-7| - 1 = 1\). Alphabet gap = \(|16-18| - 1 = 1\). Match! (P-R)
• P (16) at pos 5 and A (1) at pos 8: Word gap = \(|5-8| - 1 = 2\). Alphabet gap = \(|16-1| - 1 = 14\). No match.
• P (16) at pos 5 and T (20) at pos 9: Word gap = \(|5-9| - 1 = 3\). Alphabet gap = \(|16-20| - 1 = 3\). Match! (P-T)

Checking from the sixth letter (L at pos 6):

• L (12) at pos 6 and R (18) at pos 7: Word gap = \(|6-7| - 1 = 0\). Alphabet gap = \(|12-18| - 1 = 5\). No match.
• L (12) at pos 6 and A (1) at pos 8: Word gap = \(|6-8| - 1 = 1\). Alphabet gap = \(|12-1| - 1 = 10\). No match.
• L (12) at pos 6 and T (20) at pos 9: Word gap = \(|6-9| - 1 = 2\). Alphabet gap = \(|12-20| - 1 = 7\). No match.

Checking from the seventh letter (R at pos 7):

• R (18) at pos 7 and A (1) at pos 8: Word gap = \(|7-8| - 1 = 0\). Alphabet gap = \(|18-1| - 1 = 16\). No match.
• R (18) at pos 7 and T (20) at pos 9: Word gap = \(|7-9| - 1 = 1\). Alphabet gap = \(|18-20| - 1 = 1\). Match! (R-T)

Checking from the eighth letter (A at pos 8):

• A (1) at pos 8 and T (20) at pos 9: Word gap = \(|8-9| - 1 = 0\). Alphabet gap = \(|1-20| - 1 = 18\). No match.

We check in both forward and backward directions, but count each unique pair only once (e.g., P-R is the same pair as R-P). The pairs found are:

1. I-L (from pos 3 to 6, or 6 to 3)
2. U-R (from pos 4 to 7, or 7 to 4)
3. P-R (from pos 5 to 7, or 7 to 5)
4. P-T (from pos 5 to 9, or 9 to 5)
5. R-T (from pos 7 to 9, or 9 to 7)

There are 5 such pairs of letters in the new word "SIIUPLRAT".

Summary of Pair Calculation

Let's confirm the count of pairs where the number of letters between them in the word equals the number of letters between them in the alphabet:

All 5 pairs found satisfy the condition.

Therefore, there are 5 pairs of letters in the new word that have as many letters between them as they have in the English alphabet series.

Revision Table: Key Concepts

Additional Information: Word Reasoning Puzzles

Word reasoning puzzles often involve manipulating letters or words based on specific rules. Common types include:

• Letter arrangement or rearrangement.
• Coding and decoding based on letter positions or patterns.
• Finding pairs of letters with specific gaps (like in this problem).
• Sequence or series based on letter properties.

To solve these puzzles effectively, it is helpful to:

• Clearly understand the rules provided.
• Systematically apply the rules step-by-step.
• Write down intermediate steps (like the word after each interchange).
• Know the alphabetical position of each letter.
• Be careful with counting gaps (excluding the letters themselves).
• Check pairs in both forward and backward directions unless specified otherwise.

These types of questions test logical reasoning, attention to detail, and understanding of basic letter sequencing.