Four groups of numbers are given out of which the numbers in three groups bear…

Concept

In a 'find the differently related group' question, three of the four number-triplets are built by one shared rule while the fourth follows it only partway. Here the shared construction is: start from a perfect square n2, then add (n+1)2 + 1, then (n+2)2 + 2 - three consecutive squares with offsets 0, 1, 2. Checking every group against this template, and cross-checking with the differences between consecutive terms, shows exactly which group departs from it.

Applying the template to each group

1. (169, 197, 227): 132 = 169 (offset 0); 142 + 1 = 196 + 1 = 197 (offset 1); 152 + 2 = 225 + 2 = 227 (offset 2) - this triplet fits the n2, (n+1)2+1, (n+2)2+2 template exactly.

2. (289, 325, 363): 172 = 289 (offset 0); 182 + 1 = 324 + 1 = 325 (offset 1); 192 + 2 = 361 + 2 = 363 (offset 2) - this triplet also fits the template exactly.

3. (81, 101, 123): 92 = 81 (offset 0); 102 + 1 = 100 + 1 = 101 (offset 1); 112 + 2 = 121 + 2 = 123 (offset 2) - this triplet fits the template exactly too.

4. (121, 145, 197): 112 = 121 (offset 0) and 122 + 1 = 144 + 1 = 145 (offset 1) both fit, but the third term should then be 132 + 2 = 169 + 2 = 171, not 197. The actual third term, 197, instead equals 142 + 1 - reusing the offset-1 rule on a skipped base rather than continuing to offset 2. This triplet breaks the template at its third term.

Cross-check

This is confirmed by successive differences: for a conforming triplet, the gap between the 1st and 2nd terms is 2n + 2 and the gap between the 2nd and 3rd is 2n + 4, so the second difference (the gap between the gaps) is always 2. (169, 197, 227): gaps 28 then 30, second difference 2. (289, 325, 363): gaps 36 then 38, second difference 2. (81, 101, 123): gaps 20 then 22, second difference 2. (121, 145, 197): gaps 24 then 52, second difference 28 - far from the constant 2 the other three share.

Answer

So (121, 145, 197) is the group whose numbers are differently related.