What should come in place of question mark (?) in the following ?
[Figure: three four-petal (crossed-loop) diagrams, each with four outer numbers and one number in the centre. Figure 1 — top-left 16,

Question

What should come in place of question mark (?) in the following ?
[Figure: three four-petal (crossed-loop) diagrams, each with four outer numbers and one number in the centre. Figure 1 — top-left 16, top-right 21, bottom-left 15, bottom-right 10, centre 36. Figure 2 — top-left 7, top-right 10, bottom-left 13, bottom-right 10, centre 16. Figure 3 — top-left 15, top-right ?, bottom-left 21, bottom-right 14, centre 64.]

UnlockIAS · Accepted Answer

Correct — B, 22. The rule is: read the two petal pairs in one fixed direction, and the centre is (that difference + 1) squared. Fix the direction as top-right minus top-left for the upper pair and bottom-left minus bottom-right for the lower pair — the two must give the same number. Figure 1: 21 − 16 = 5 and 15 − 10 = 5, so the difference is 5 and the centre should be (5 + 1)² = 36, which is exactly what is printed. Figure 2: 10 − 7 = 3 and 13 − 10 = 3, so the centre should be (3 + 1)² = 16 — again exactly what is printed. Figure 3 hands you the centre instead of the petal: 64 = 8², so difference + 1 = 8 and the difference is 7. Check it against the pair you can see: 21 − 14 = 7. ✓ Now apply it upward: ? − 15 = 7, so ? = 22. The same value falls out of an equivalent reading — in every figure the two left-hand petals sum to the two right-hand petals (16 + 15 = 31 = 21 + 10; 7 + 13 = 20 = 10 + 10), and 15 + 21 = 36 = ? + 14 gives ? = 22 again. The two rules are algebraically the same condition, which is why they agree, but the squared-centre version is the stronger one to trust because it uses the centre numbers as well: it reproduces all three centres from the petals alone, and the two complete figures each supply the difference twice over. That is five agreeing constraints across fourteen printed numbers — far too many for a rule that had merely been fitted to one diagram.

Why the other options are wrong — (a) 18 − 15 = 3, which would make the centre (3 + 1)² = 16. That is Figure 2's centre, not Figure 3's, and it also contradicts the lower pair of the same diagram, where 21 − 14 = 7. Every wrong option here is the value that would import another figure's centre into this one.  (c) 20 − 15 = 5, which would give a centre of (5 + 1)² = 36 — Figure 1's centre. The printed centre is 64, and the bottom pair independently fixes the difference at 7, so 5 cannot be right. This is the closest miss and the option most students settle on.  (d) 36 is simply Figure 1's centre value offered back as a petal. It would make the top difference 36 − 15 = 21 against a bottom difference of 21 − 14 = 7, breaking the equality that both complete diagrams obey, and it would demand a centre of (21 + 1)² = 484.

Concept: A figure-pattern question gives you two or three completed diagrams and one with a gap. The completed ones are not decoration — they are the only evidence for the rule, and a rule is only usable when it reproduces every number in every completed diagram, not just the one you happened to notice. The safe method is therefore fixed: (1) write out the numbers in a consistent positional scheme, such as top-left, top-right, bottom-left, bottom-right, centre; (2) look for an operation that turns the outer numbers into the centre; (3) test it on the second complete figure before touching the third; (4) only then solve for the gap; (5) confirm with a second, independent relation if the diagram offers one. Perfect squares in a centre position — 36, 16, 64 here — are a strong hint, because a setter who wants the centre to encode something small usually squares it.

State PCS 2024 Q130 — Science & Technology PYQ with Answer

State PCS 2024 Q130 — Science & Technology PYQ with Answer