Decoding Letter Cluster Transformations

This question involves transforming given letter clusters based on specific rules for vowels and consonants and then identifying how many transformed clusters contain no vowels.

Understanding the Transformation Rules

The core task is to apply two distinct rules to each letter in the provided words (OPE, HER, KID, SKY):

• Vowel Rule: Each vowel (A, E, I, O, U) is replaced by the letter that comes immediately after it in the English alphabet. Mathematically, if $V$ is a vowel, its transformation is $ V \rightarrow V+1 $.
• Consonant Rule: Each consonant is replaced by the letter that comes immediately before it in the English alphabet. Mathematically, if $C$ is a consonant, its transformation is $ C \rightarrow C-1 $.

After applying these rules to all letters in a word, we need to check the resulting cluster for the presence of vowels.

Analyzing Each Letter Cluster

Let's transform each word step-by-step:

1. Word: OPE

Applying the rules:

• 'O' is a vowel. It changes to the next letter: $ O \rightarrow P $.
• 'P' is a consonant. It changes to the previous letter: $ P \rightarrow O $.
• 'E' is a vowel. It changes to the next letter: $ E \rightarrow F $.

The transformed cluster for OPE is POF.

Checking for vowels in POF: The letter 'O' is present, which is a vowel. Therefore, this cluster does not meet the condition of having no vowels.

2. Word: HER

Applying the rules:

• 'H' is a consonant. It changes to the previous letter: $ H \rightarrow G $.
• 'E' is a vowel. It changes to the next letter: $ E \rightarrow F $.
• 'R' is a consonant. It changes to the previous letter: $ R \rightarrow Q $.

The transformed cluster for HER is GFQ.

Checking for vowels in GFQ: The letters are G, F, Q. None of these are vowels (A, E, I, O, U). Therefore, this cluster meets the condition of having no vowels.

3. Word: KID

Applying the rules:

• 'K' is a consonant. It changes to the previous letter: $ K \rightarrow J $.
• 'I' is a vowel. It changes to the next letter: $ I \rightarrow J $.
• 'D' is a consonant. It changes to the previous letter: $ D \rightarrow C $.

The transformed cluster for KID is JJC.

Checking for vowels in JJC: The letters are J, J, C. None of these are vowels. Therefore, this cluster meets the condition of having no vowels.

4. Word: SKY

Applying the rules:

• 'S' is a consonant. It changes to the previous letter: $ S \rightarrow R $.
• 'K' is a consonant. It changes to the previous letter: $ K \rightarrow J $.
• 'Y' is typically treated as a consonant in such problems. It changes to the previous letter: $ Y \rightarrow X $.

The transformed cluster for SKY is RJX.

Checking for vowels in RJX: The letters are R, J, X. None of these are vowels. Therefore, this cluster meets the condition of having no vowels.

Counting Letter-Clusters Without Vowels

After transforming each word and checking the resulting letter-cluster, we found the following:

• OPE transformed to POF (Contains vowel 'O')
• HER transformed to GFQ (Contains no vowels)
• KID transformed to JJC (Contains no vowels)
• SKY transformed to RJX (Contains no vowels)

The letter-clusters that will have no vowel after the transformation are GFQ (from HER), JJC (from KID), and RJX (from SKY).

Counting these, we find there are 3 such letter-clusters.