Non-Verbal Reasoning for CSAT

Non-verbal reasoning is problem-solving with figures, patterns and numbers instead of words: continue a series, find the missing number in a matrix, spot the odd one out, or complete an analogy. UPSC's favourite form is the 'missing character' matrix, asked almost every year. Because Paper II is qualifying at 33 percent, this is a scoring topic: the patterns come from a small, learnable family, and each question is a self-contained puzzle.

The non-verbal family

Non-verbal questions fall into five types. A series asks for the next or missing term in a sequence. A missing-character item hides one entry in a grid or set of figures governed by a rule. An analogy gives A : B :: C : ? and asks for the parallel term. Classification (odd one out) asks which item does not belong. Figure completion asks which piece finishes a pattern. In CSAT, number-based forms dominate: number series and missing-character matrices carry the topic.

Number series: the pattern checklist

Attack every series with the same checklist. First, differences between consecutive terms: constant (arithmetic), growing steadily (squares), or doubling (geometric). Second, ratios: each term a multiple of the previous. Third, famous forms: squares, cubes, and their neighbours (n^2 plus or minus 1). Fourth, alternating or interleaved patterns: treat odd and even positions as two separate series. Fifth, digit operations: sums or products of digits driving the next term. Worked example: 2, 6, 12, 20, 30, ?. Differences are 4, 6, 8, 10, growing by 2 each step, so the next difference is 12 and the missing term is 42.

Missing character: the matrix method

For grids, test rules in this order: along rows first, then columns, then diagonals, then special positions (corners versus centre). Common rules are addition or subtraction across a row, multiplication with a constant offset, digit sums, and squares. Worked example [UPSC CSE 2021]: the matrix 7B 10A 3C / 3C 9B 6A / 10A 13C ?. Row-wise, the numbers satisfy first plus third equals second (7+3=10; 3+6=9; 10+?=13, so ? = 3), while the letters' positions (B=2, A=1, C=3) sum to 6 in each row (2+1+3; 3+2+1; 1+3+?=6, so ? = B). The missing entry is 3B. Note the double rule: numbers and letters were checked independently.

Worked example [UPSC CSE 2023]: 'If 7*9*10 = 8, 9*11*30 = 5, 11*17*21 = 13, what is 23*4*15?' Adding gives 26, 50, 49, then summing the digits gives 8, 5, 13: the rule is add the three numbers, then add the digits of the total. For 23*4*15: 23+4+15 = 42, and 4+2 = 6. The digit-sum collapse is a favourite UPSC move: when totals look messy, reduce them.

Sequences with a changing multiplier

Some series hide the pattern in the multipliers between terms rather than the terms themselves. Worked example [UPSC CSE 2021]: Sequence-I is 8, 4, 6, 15, 52.5, 236.25 and Sequence-II is 5, A, B, C, D, E with identical structure. The multipliers in Sequence-I are 0.5, 1.5, 2.5, 3.5, 4.5, an arithmetic progression with difference 1. Applying the same multipliers from 5: A = 2.5, B = 3.75, C = 9.375. Whenever raw terms look irregular, write the ratios or differences between them; the pattern usually lives there.

Odd one out and analogies

Classification questions need exactly one rule that fits all but one item; the commonest rules are perfect squares or cubes, primes, and digit properties. Worked example: among 121, 144, 169 and 180, the first three are 11^2, 12^2 and 13^2, while 180 is not a perfect square, so 180 is the odd one out. Analogies need the precise relation: 3 : 27 works as 'cube of', giving 4 : 64, but check alternatives ('times 9' would give 36), because UPSC sometimes plants two plausible relations and only one survives the options.

Common traps

Five traps recur. One, stopping at the first pattern that fits the given terms but fails the options: always verify against all terms. Two, checking rows only in a matrix and missing a column rule (or vice versa). Three, ignoring letters in alphanumeric entries: they follow their own rule using alphabetical positions. Four, in series, forgetting interleaved patterns: if no single rule fits, split odd and even positions. Five, overcomplicating: UPSC rules are usually one operation (add, multiply, digit sum), not three nested ones.

Speed tips

Three habits help. First, write differences or ratios in the margin immediately; most series surrender within seconds. Second, in matrices, check totals first: if every complete row sums to the same number, you are done. Third, memorise the small arsenal cold: squares to 20^2, cubes to 10^3, primes to 50, and powers of 2 to 2^10, so recognition is instant rather than computed.

Key Terms

• Number series: a sequence of numbers following a hidden rule; the task is to find the next or missing term.
• Missing character: a grid or figure set with one entry hidden; the entry is found from the rule governing rows, columns or positions.
• Matrix: a rectangular grid of numbers (sometimes with letters) in which each row or column follows the same arithmetic rule.
• Analogy: a proportion of the form A : B :: C : ?, asking for the term related to C exactly as B is related to A.
• Classification (odd one out): a set of items in which all but one share a property; the task is to identify the exception.
• Alphanumeric series: entries mixing numbers and letters, where each part follows its own rule (letters use alphabetical positions).
• Common difference: the constant gap between consecutive terms of an arithmetic progression; a steadily growing gap hints at squares.
• Geometric progression: a sequence in which each term is a constant multiple of the previous one (e.g. doubling each step).

Practice questions