In a certain code language, the word 'TEARS' is written as 'UGDVX'. Following the same sequential letter-shift pattern, how would the word 'SMILE' be written?

Introduction / Context:
This coding question uses varying forward shifts in the alphabet for each successive letter. TEARS is coded as UGDVX, and we must infer the pattern of shifts and then apply it to the word SMILE. This problem checks our skill in detecting a changing letter shift (not the same for all positions) and carefully applying it to another word of similar length.

Given Data / Assumptions:

• TEARS becomes UGDVX in the code language.
• The number of letters remains the same; only their alphabetical positions change.
• The amount of shift between original and coded letters can vary by position.
• The same position-wise shifts must be applied to SMILE.

Concept / Approach:
We compare each letter of TEARS with its coded counterpart in UGDVX. By computing the forward shift for each position, we can see whether there is a pattern such as +1, +2, +3 and so on. Once we know the exact sequence of shifts, we reuse it on the word SMILE, shifting each letter by the same amount as the corresponding position in TEARS.

Step-by-Step Solution:
Step 1: Align the words by position: T E A R S and U G D V X. Step 2: Compute the shift for each position: T -> U: T (20th letter) to U (21st) is +1. E -> G: E (5th) to G (7th) is +2. A -> D: A (1st) to D (4th) is +3. R -> V: R (18th) to V (22nd) is +4. S -> X: S (19th) to X (24th) is +5. Step 3: Therefore, the pattern of shifts is +1, +2, +3, +4, +5 for positions 1 to 5. Step 4: Apply the same pattern to SMILE (S, M, I, L, E). Step 5: For position 1: S (19th) +1 = T (20th). Step 6: For position 2: M (13th) +2 = O (15th). Step 7: For position 3: I (9th) +3 = L (12th). Step 8: For position 4: L (12th) +4 = P (16th). Step 9: For position 5: E (5th) +5 = J (10th). Step 10: Joining these gives TOLPJ as the code for SMILE.

Verification / Alternative check:
We can quickly reapply the pattern to TEARS to confirm: T(+1) = U, E(+2) = G, A(+3) = D, R(+4) = V, S(+5) = X. Since this works perfectly, and our application of the same pattern to SMILE produced TOLPJ, we are confident that TOLPJ is correct.

Why Other Options Are Wrong:
Option B: TOLNG alters the shifts at later positions and does not maintain the +3, +4, +5 pattern correctly. Option C: TOJPJ changes the middle letter incorrectly. Option D: TOKNH significantly deviates from the established shift sequence. Option E: TNLPJ changes multiple letters without any consistent relationship to the rule derived from TEARS.

Common Pitfalls:
A frequent mistake is assuming a constant shift for all letters rather than observing that the shift increases by 1 at each step. Another error is miscounting alphabet positions, especially near the beginning or end of the alphabet. Writing out the alphabet with indices or mentally verifying positions prevents these errors.

Final Answer:
According to the given coding rule, SMILE is written as TOLPJ.