In a certain code language, the word "TRUMPET" is written as "7591427" and the word "SORROW" is written as "385586". Using the same letter to digit code, how is the word "EMPRESS" written in that code language?

Introduction:
This coding and decoding question deals with a substitution cipher in which each letter of the alphabet is replaced by a specific digit. We are given two coded words, "TRUMPET" and "SORROW", along with their digit sequences. Our task is to determine the digit string corresponding to the word "EMPRESS" by extracting and using a consistent set of letter to digit mappings.

Given Data / Assumptions:
The mappings are provided through two examples: TRUMPET is coded as 7591427 and SORROW is coded as 385586. We assume that the code is consistent across all words. That means each letter always maps to the same digit, and no two different letters share the same digit within this question. The word to be encoded is "EMPRESS", which contains letters E, M, P, R, and S, all of which appear in the given examples.

Concept / Approach:
The approach is to align letters and digits position wise in the given coded words and then build a mapping table. Once we know the digit for each relevant letter, the encoding of "EMPRESS" is a straightforward lookup operation. Consistency is crucial: every appearance of a letter must be associated with the same digit in all examples.

Step-by-Step Solution:
Step 1: Consider "TRUMPET" coded as "7591427". Match letters to digits: T corresponds to 7, R to 5, U to 9, M to 1, P to 4, E to 2, and again T to 7 which confirms T = 7.Step 2: Now consider "SORROW" coded as "385586". Match letters: S corresponds to 3, O to 8, R to 5, R again to 5 confirming R = 5, O again to 8 confirming O = 8, and W to 6.Step 3: From these examples, we have the following mappings: T = 7, R = 5, U = 9, M = 1, P = 4, E = 2, S = 3, O = 8, W = 6.Step 4: The word "EMPRESS" has letters E, M, P, R, E, S, S in that order.Step 5: Using the mapping, we get E = 2, M = 1, P = 4, R = 5, E again = 2, S = 3, and S again = 3.Step 6: Therefore, the code for "EMPRESS" is 2 1 4 5 2 3 3, which is 2145233.

Verification / Alternative check:
We can verify our mapping by substituting the digits back into the original coded examples. For "TRUMPET", T = 7, R = 5, U = 9, M = 1, P = 4, E = 2, T = 7 gives 7591427, matching the given code. For "SORROW", S = 3, O = 8, R = 5, R = 5, O = 8, W = 6 gives 385586, again matching perfectly.

Why Other Options Are Wrong:
Option 2145237 incorrectly changes the final digit to 7, which would correspond to T rather than S. Option 2154323 reorders digits in a way that would require inconsistent letter mappings. Option 3154233 assigns the first digit 3 to E, which contradicts E = 2. Option 2154233 misaligns the middle digits and again conflicts with the established mapping for M and P.

Common Pitfalls:
Candidates sometimes mix up digits when transferring them from the mapping table to the target word or fail to distinguish between similar looking letters. Another mistake is to forget to check for consistency across both given coded words, which can lead to incorrect interpretations of the mapping.

Final Answer:
The word "EMPRESS" is written as 2145233 in the given code language.