Decoding the 'FAN' to '63' Logic

The problem presents a coding challenge where words are represented by numbers. We need to find the pattern used to encode 'FAN' as '63' and 'DEAR' as '112', and then apply it to 'BABY'. Let's break down the logic.

Mapping Letters to Alphabetical Positions

First, we assign numerical values to each letter based on its position in the English alphabet:

• A = 1, B = 2, C = 3, ..., Z = 26

Analyzing the Code for 'FAN'

Let's find the sum of the alphabetical positions for the letters in 'FAN':

• F corresponds to 6
• A corresponds to 1
• N corresponds to 14

The sum of these values is: $6 + 1 + 14 = 21$

Now, we need to relate this sum (21) to the given code (63). We observe that $63 \div 21 = 3$. The number '3' is exactly the number of letters in the word 'FAN'. This suggests the pattern might be: (Sum of letter values) $\times$ (Number of letters).

Let's test this hypothesis: $21 \times 3 = 63$. This matches the given code.

Verifying the Pattern with 'DEAR'

Let's apply the same potential pattern to the word 'DEAR':

• D corresponds to 4
• E corresponds to 5
• A corresponds to 1
• R corresponds to 18

The sum of these values is: $4 + 5 + 1 + 18 = 28$

'DEAR' has 4 letters. According to our hypothesized pattern, the code should be the sum multiplied by the number of letters:

Calculation: $28 \times 4 = 112$.

This matches the given code for 'DEAR'. Our pattern is confirmed.

Applying the Pattern to 'BABY'

Now, we use the established pattern to find the code for 'BABY':

1. Find the sum of the alphabetical positions of the letters in 'BABY':
   • B corresponds to 2
   • A corresponds to 1
   • B corresponds to 2
   • Y corresponds to 25
   The sum is: $2 + 1 + 2 + 25 = 30$.
2. Determine the number of letters in 'BABY'. The word 'BABY' has 4 letters.
3. Apply the pattern: (Sum of letter values) $\times$ (Number of letters). Calculation: $30 \times 4 = 120$.

Therefore, 'BABY' will be written as '120' in this code language.