BPSC 69th CCE Prelims General Studies · Question 146 of 144

Correct — B, 77. The booklet prints these nine numbers as a three-by-three block, and that layout is the question: 44 49 37 in the first row, 52 ? 41 in the second, 58 35 53 in the third. Two rows are complete, so a candidate rule can be tested twice before it is used once. Take the two outer numbers of a row and subtract, then multiply by 7. Row 1: 44 − 37 = 7, and 7 × 7 = 49, which is the printed middle entry. Row 3: 58 − 53 = 5, and 5 × 7 = 35, again the printed middle entry. The same rule applied to row 2 gives 52 − 41 = 11, and 11 × 7 = 77, which is option (b). Two independent confirmations before the rule is applied is the whole discipline of a grid item, and it also supplies a free filter: every middle entry the paper prints is a multiple of 7, so any option that is not divisible by 7 can be struck out without doing the subtraction at all. That alone removes 66. Of the three multiples of 7 left — 56, 63 and 77 — only 77 corresponds to the row-2 difference of 11, since 56 needs a difference of 8 and 63 a difference of 9, and neither appears anywhere in row 2.

• (a)56 — 56 is 7 × 8, so it belongs to the right family and looks reassuring. The 8 comes from subtracting down the left-hand column, 52 − 44, instead of across the two ends of row 2. The rule established on rows 1 and 3 runs along a row, not down a column, and along row 2 the difference is 52 − 41 = 11.
• (c)66 — 66 is not a multiple of 7 and can be eliminated in a second, because both printed middle entries — 49 and 35 — are multiples of 7. It is what you get from 6 × 11: the correct row-2 difference of 11 with the multiplier read off as 6 rather than 7, a slip that comes from checking the rule on only one row.
• (d)63 — 63 is 7 × 9, again the right family with the wrong difference. A single transcription slip in the subtraction — reading 52 − 41 as 9 — produces it. It is also the answer a candidate reaches if they average the two known differences, 7 and 5, get 6 and then drift; nothing in the grid supports interpolating between rows, because the rule operates inside each row separately.

A missing-number grid is a series question folded into two dimensions. Nine or twelve numbers are laid out in a block and one cell is blank; the rule may run along the rows, down the columns, around the perimeter, or between the two diagonals, and the examiner supplies at least two complete lines so that the rule can be established before it is applied. The operations used are deliberately elementary — add, subtract, multiply, divide, or a difference or a sum scaled by a constant — because the item is testing whether a candidate searches systematically rather than whether they can compute. The systematic search has a fixed order: try the rows first, since most grids are row-based; if no row rule fits both complete rows, try the columns; only then look at diagonals or at the block as a whole. A rule is accepted only when it reproduces every complete line, never on the strength of the first line alone.

The reasoning here has two stages, and skipping the first is what costs the mark. Stage one is to notice that the nine numbers are a grid and not a sequence — read as a single line, 44 49 37 52 ? 41 58 35 53 has no pattern, and a candidate who keeps trying to force one runs out of time. Re-block them into threes and the structure appears immediately. Stage two is to find the rule on the two intact rows and confirm it on both before touching the blank. Row 1 alone is a trap: 44 − 37 = 7 and the middle entry is 49, which is also 7², so the squaring rule fits row 1 perfectly. Row 3 refutes it — 58 − 53 = 5 and 5² is 25, not the printed 35 — and only the multiply-by-7 reading survives both rows. Had the squaring rule been adopted, row 2 would have produced 11² = 121, which does not appear among the options at all; when your rule yields a number that is not on the list, that is a signal to go back to the rows, not to pick the nearest option. The discriminating fact is simply that both printed middle entries are seven times the outer difference of their own row.

• The nine numbers are printed as a 3 × 3 block: 44 49 37 in row 1, 52 ? 41 in row 2, 58 35 53 in row 3
• Rule confirmed on both complete rows — (left number − right number) × 7 = middle number: row 1 gives (44 − 37) × 7 = 49, row 3 gives (58 − 53) × 7 = 35
• Applied to row 2 the rule gives (52 − 41) × 7 = 11 × 7 = 77
• Every middle entry is a multiple of 7, so 66 fails a divisibility test before any subtraction is done; 56 = 7 × 8 and 63 = 7 × 9 need differences of 8 and 9, neither of which occurs in row 2
• The squaring near-miss: 49 = 7² fits row 1, but row 3 would then require 5² = 25 against a printed 35 — which is why a rule must be tested on every complete line

• Reading the nine numbers as a single sequence instead of re-blocking them into a 3 × 3 grid
• Fixing the rule from the first complete row alone — here row 1 also fits 'square the difference', and row 3 is what refutes it
• Mixing a column difference into a row rule, which is exactly what produces 56

BPSC has used the missing-number grid in successive editions and keeps the same architecture each time — a 3 × 3 block, two complete rows to establish the rule and one blank cell — changing only the arithmetic operation, a scaled difference in 2023 and a scaled ratio in 2025. UPSC places this family in CSAT Paper II under general mental ability, usually as a figure or letter matrix rather than a purely numeric one.

No directly related past PYQ was found.

• practice — not a real PYQ

  Select the missing number from the given alternatives : 30 42 23 / 41 ? 26 / 55 30 50

  • (a)84
  • (b)90
  • (c)96
  • (d)88

  Answer(b) 90 — in each row (left − right) × 6 = middle: row 1 gives (30 − 23) × 6 = 42 and row 3 gives (55 − 50) × 6 = 30, so row 2 gives (41 − 26) × 6 = 90.

• practice — not a real PYQ

  In the grid below the rule runs along each row. Select the missing number : 12 15 27 / 18 21 39 / 25 ? 56

  • (a)29
  • (b)30
  • (c)31
  • (d)33

  Answer(c) 31 — the third entry in each row is the sum of the first two (12 + 15 = 27, 18 + 21 = 39), so the missing number is 56 − 25 = 31.