In a certain coded language, 'FATE' is coded as '4265' and 'FALE' is coded as…

Concept: In letter-to-number coding puzzles, when two coded words share some letters, the digits coding those shared letters are also common to both numeric codes (though not necessarily in the same position). A letter that appears in only one of the two words is coded by the one digit left over in that word's code after every shared digit has been removed.

1. Compare the letters of 'FATE' and 'FALE': both words share the letters F, A, and E; only 'T' belongs to 'FATE' alone and only 'L' belongs to 'FALE' alone.

2. Compare the digits of '4265' and '2145': both codes share the digits 4, 2, and 5 (listed in different order), so the shared letters {F, A, E} correspond to the shared digits {4, 2, 5} as a set.

3. In '4265' (the code for FATE), the digit left after removing 4, 2, and 5 is 6 — this is the code for the letter unique to FATE, i.e. T.

4. In '2145' (the code for FALE), the digit left after removing 4, 2, and 5 is 1 — this is the code for the letter unique to FALE, i.e. L.

Cross-check: As a check, reassembling the digits confirms the mapping: {4, 2, 5} for {F, A, E} plus 6 for T reproduces '4265' as a rearrangement of FATE's letters, and {4, 2, 5} for {F, A, E} plus 1 for L reproduces '2145' as a rearrangement of FALE's letters — both given codes are fully accounted for.

Therefore, the code for 'L' is 1.