Given that in a code language, ‘645’ means ‘day is warm’; ‘42’ means ‘warm…

Concept: In sentence-substitution coding each word of a message is replaced by one fixed digit, and the digits are printed in a scrambled order, so position carries no information. Two consequences drive every item of this kind: a word shared by two messages must be encoded by a digit shared by their two codes, and a word that occurs in only one message must be encoded by a digit that occurs in only that one code.

Application: Apply the second consequence to the word the question asks about.

1. The word ‘sunny’ occurs in only one of the three statements — ‘spring is sunny’, coded 634 — so the digit for ‘sunny’ is one of 6, 3 and 4, and it must not turn up in any other code.

2. The digit 6 also stands inside 645, the code for ‘day is warm’. If 6 encoded ‘sunny’, then ‘sunny’ would have to be one of the words of ‘day is warm’, which it is not, so 6 is eliminated.

3. The digit 4 also stands inside 645 (‘day is warm’) and inside 42 (‘warm spring’); the same argument eliminates 4.

4. The digit 3 stands inside 634 and inside no other code, so 3 is the digit that encodes ‘sunny’.

Cross-check: The substitution route reaches the same mapping. ‘spring’ is common to ‘warm spring’ (42) and ‘spring is sunny’ (634), and 4 is the single digit common to {4, 2} and {6, 3, 4}, so spring = 4. ‘is’ is common to ‘day is warm’ (645) and ‘spring is sunny’ (634); their codes share 6 and 4, and 4 is already committed to ‘spring’, so is = 6. Inside 634 that leaves 3 for ‘sunny’.

Result: The digit 3 represents ‘sunny’.

Note: One caution about the printed data: ‘warm’ is common to 645 and 42, whose codes share only the digit 4, yet 4 has already been fixed to ‘spring’, so the three statements do not admit a perfectly consistent one-word-one-digit mapping for ‘warm’. That flaw does not touch ‘sunny’ — both routes above fix it to 3 independently, and the official UGC NET June 2012 Paper I answer key also marks 3.