Analogy Analysis: CREAM to FPJYZ

This problem asks us to identify a coding pattern between the first pair of words, CREAM and FPJYZ, and then apply that same pattern to the word CHECK to find the missing corresponding word.

We will analyze the relationship by looking at the positions of the letters in the English alphabet (A=1, B=2, ..., Z=26).

Decoding the Transformation Pattern

Let's break down the transformation from CREAM to FPJYZ letter by letter:

• C is the $3^{rd}$ letter. It transforms into F, the $6^{th}$ letter. The shift is $6 - 3 = +3$.
• R is the $18^{th}$ letter. It transforms into P, the $16^{th}$ letter. The shift is $16 - 18 = -2$.
• E is the $5^{th}$ letter. It transforms into J, the $10^{th}$ letter. The shift is $10 - 5 = +5$.
• A is the $1^{st}$ letter. It transforms into Y, the $25^{th}$ letter. The shift is $25 - 1 = 24$. Alternatively, thinking cyclically, moving back 2 positions from A lands on Y ($1 \rightarrow Z \rightarrow Y$). This is a shift of $-2$. ($1 - 2 = -1$. In a 26-letter cycle, $-1$ corresponds to $25$).
• M is the $13^{th}$ letter. It transforms into Z, the $26^{th}$ letter. The shift is $26 - 13 = +13$.

The pattern of shifts observed is: +3, -2, +5, -2, +13.

Let's examine this pattern more closely:

• The shifts for the first four letters seem fixed: +3, -2, +5, -2.
• The shift for the fifth letter (M) is +13. Notice that M is the $13^{th}$ letter of the alphabet. This suggests the final shift might depend on the letter itself.

Hypothesized Pattern: The shifts applied are +3, -2, +5, -2, and +N, where N is the alphabetical position of the 5th letter of the original word.

Applying the Pattern to CHECK

Now, let's apply this discovered pattern to the word CHECK.

• C (the $3^{rd}$ letter): Apply the first shift (+3). $3 + 3 = 6$. The $6^{th}$ letter is F.
• H (the $8^{th}$ letter): Apply the second shift (-2). $8 - 2 = 6$. The $6^{th}$ letter is F.
• E (the $5^{th}$ letter): Apply the third shift (+5). $5 + 5 = 10$. The $10^{th}$ letter is J.
• C (the $3^{rd}$ letter): Apply the fourth shift (-2). $3 - 2 = 1$. The $1^{st}$ letter is A.
• K (the $11^{th}$ letter): Apply the fifth shift. According to our hypothesis, the shift should be the alphabetical position of K, which is 11. So, the shift is +11. $11 + 11 = 22$. The $22^{nd}$ letter is V.

Combining these transformed letters, we get FFJAV.

Identifying the Correct Option

Comparing our result FFJAV with the given options:

• Option 1: FZJFV
• Option 2: FFJAV
• Option 3: FPJZV
• Option 4: FFJFV

Our derived word FFJAV matches Option 2.