In a certain code language, the word "CUTE" is written as "9251" and the word "REST" is written as "4135". Using the same letter to digit code, how is the word "RESCUE" written?

Introduction / Context:
This is a classic coding decoding question where each letter of the alphabet is represented by a specific digit. You are given the coded forms of two words and asked to find the code for a third related word. The challenge lies in identifying a consistent mapping between letters and digits and then using that mapping to encode the new word correctly.

Given Data / Assumptions:
- "CUTE" is coded as "9251".
- "REST" is coded as "4135".
- The required word is "RESCUE".
- The same letter always has the same digit, and the same digit always represents the same letter within this code language.

Concept / Approach:
The approach is to align each letter of the given words with its corresponding digit position. Once we know which digit belongs to each letter, we can simply substitute letters of the new word with their matching digits. No mathematical operations are involved here; it is purely a matter of consistent letter digit substitution across all examples.

Step-by-Step Solution:
Step 1: From "CUTE" → "9251", assign letter to digit mapping by position. Step 2: C → 9, U → 2, T → 5, E → 1. Step 3: From "REST" → "4135", we get R → 4, E → 1, S → 3, T → 5. Step 4: Combine the information from both words to obtain the full mapping: R → 4, E → 1, S → 3, C → 9, U → 2, T → 5. Step 5: Now encode "RESCUE" letter by letter using this mapping. Step 6: R → 4, E → 1, S → 3, C → 9, U → 2, E → 1, so "RESCUE" is written as 413921.

Verification / Alternative check:
We can verify that this mapping is consistent by reconstructing the original codes. Plugging the mapping back into "CUTE" gives C(9) U(2) T(5) E(1) which forms "9251". For "REST", we get R(4) E(1) S(3) T(5) which forms "4135". Because the mapping reproduces both original codes perfectly, it is reliable. Therefore, applying the same mapping to "RESCUE" must give the correct encoded form 413921.

Why Other Options Are Wrong:
- Option 431291 changes the positions of digits and does not match the letter order in "RESCUE".
- Option 423911 assigns wrong digits for one or more letters and breaks the established mapping.
- Option 413912 swaps the final two digits, which would imply U and E share or swap codes, contradicting the consistent mapping deduced earlier.

Common Pitfalls:
A common mistake is to treat the numbers as if they must form a recognizable pattern or to try to perform arithmetic with them. In this kind of question, the digits simply stand for letters, and the only rule is consistency. Another error is to misalign positions or assume a digit can represent more than one letter, which is not allowed in this code language.

Final Answer:
Using the same letter digit mapping, the word "RESCUE" is written in code as 413921.