If ‘367’ means ‘I am happy’; ‘748’ means ‘you are sad’ and ‘469’ means ‘happy…

Concept

In substitution (sentence) coding, every number in a code group stands for exactly one word of the sentence, and the digits are not written in the same order as the words. A single code group therefore fixes nothing on its own. The meaning of a digit is recovered by comparison: when two sentences share exactly one word and their code groups share exactly one digit, that digit must stand for that shared word.

Applying the rule to this item

The three given pairs are:

Now compare the code groups two at a time:

1. ‘367’ (I am happy) and ‘469’ (happy and sad) share exactly one word, ‘happy’, and their code groups share exactly one digit, 6. Therefore 6 = happy.

2. ‘748’ (you are sad) and ‘469’ (happy and sad) share exactly one word, ‘sad’, and their code groups share exactly one digit, 4. Therefore 4 = sad.

3. The group ‘469’ codes the three words ‘happy’, ‘and’ and ‘sad’. Two of its digits are already pinned (6 = happy and 4 = sad), so the one digit left, 9, must stand for the one word left, ‘and’.

Cross-check

• Substituting back into ‘469’: 6 = happy, 9 = and, 4 = sad — exactly the three words of ‘happy and sad’.

• ‘3’ occurs in ‘367’ alone; since 6 = happy, ‘3’ can stand only for ‘I’ or ‘am’, neither of which is ‘and’.

• ‘6’ and ‘4’ are already pinned to ‘happy’ and ‘sad’ by the two comparisons above, so neither is free to stand for ‘and’.

Hence ‘and’ is represented by 9 in this code.