Reasoning and Mental Ability — Speed Techniques for the Prelims Paper

Chapter 22: Reasoning and Mental Ability — Speed Techniques for the Prelims Paper

Reasoning is the section where preparation pays off fastest, because unlike GK, there is nothing to "know" — every question is solvable purely through method, and method can be drilled until it becomes automatic. AP Police Prelims reasoning sections cover number and letter series, coding-decoding, blood relations, direction sense, syllogism, seating arrangement, and non-verbal pattern questions, and each of these sub-types has a small number of recurring structures that repeat across papers with only the surface numbers changed. A candidate who has drilled the underlying structures can recognise a seating arrangement question as "the same shape as last week's mock" within five seconds of reading it, and that recognition is what separates a two-minute question from a six-minute one. This chapter builds elimination-first, diagram-first habits that avoid brute-force checking of every possibility — the single biggest time sink in this section.

1. Number Series — Reading the Difference Pattern First

Before attempting any arithmetic on a number series, write out the differences between consecutive terms. In the overwhelming majority of AP Police series questions, the pattern lives in the differences, not in the terms themselves.

Worked example. Find the next term: 3, 6, 11, 18, 27, 38, ?

Fast method: Differences: 6-3=3, 11-6=5, 18-11=7, 27-18=9, 38-27=11. The differences themselves form a series: 3, 5, 7, 9, 11 — increasing by 2 each time. The next difference is 13. Next term = 38+13 = 51. You never need to guess a formula for the original series; the second-level difference series is almost always simpler and often a basic arithmetic progression.

Worked example — ratio pattern. Find the next term: 5, 10, 20, 40, 80, ?

Fast method: Check ratios instead of differences when differences look irregular: 10/5=2, 20/10=2, 40/20=2, 80/40=2. Constant ratio of 2, so the next term = 80×2 = 160. The rule of thumb: try differences first; if they do not stabilise into a recognisable pattern within two or three steps, switch immediately to ratios.

Worked example — alternating operation series. Find the next term: 2, 6, 12, 20, 30, 42, ?

Fast method: Differences: 4, 6, 8, 10, 12 — increasing by 2 each time, so the next difference is 14, giving 42+14 = 56. Alternatively, recognise these as n(n+1) for n=1,2,3...: 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42, so the next term is 7×8=56. Either method reaches 56; the difference method is more reliable under time pressure because it needs no pattern recognition beyond simple subtraction.

2. Letter Series — Position-Number Conversion

Letter series questions become number series the moment you convert each letter to its alphabetical position (A=1, B=2, ... Z=26). Do this conversion first, every time, rather than trying to track letter jumps visually.

Worked example. Find the next letter: B, D, G, K, P, ?

Fast method: Convert to numbers: B=2, D=4, G=7, K=11, P=16. Differences: 2, 3, 4, 5. Next difference = 6, so next number = 16+6 = 22, which is letter V. The letter series that looked opaque on sight becomes a standard increasing-difference number series once converted.

3. Coding-Decoding Without Mental Arithmetic — the Shift-Table Method

Coding-decoding questions test whether you can spot a consistent letter-shift rule, and the fastest way to do this is to write the given word and its code one below the other, then read off the shift for just the first two or three letters — you rarely need to check every letter once the pattern is confirmed twice.

Worked example. If WATER is coded as YCVGT, what is the code for MOUSE?

Fast method: Line up W-Y, A-C, T-V, E-G, R-T. Each letter has moved forward by exactly 2 positions in the alphabet (W→Y is +2, A→C is +2, and so on — you only need to check two pairs to be confident of the rule, and a third pair confirms it beyond doubt). Apply the same +2 shift to MOUSE: M→O, O→Q, U→W, S→U, E→G. Code = OQWUG. There is no need to compute anything about "position 13 becomes position 15" mentally — once you have confirmed the shift amount, you simply move each letter forward by that many places, which is a visual alphabet-walk, not arithmetic.

Worked example — reverse-alphabet coding (a common trap format). In a certain code, A is written as Z, B as Y, C as X, and so on. How is GOLD written in this code?

Fast method: This is the "reverse alphabet" rule: the code letter's position = 27 minus the original letter's position. G is the 7th letter, so its code is the (27-7)=20th letter = T. O is the 15th letter, code = (27-15)=12th letter = L. L is the 12th letter, code = (27-12)=15th letter = O. D is the 4th letter, code = (27-4)=23rd letter = W. So GOLD is coded as TLOW. Memorise the "27 minus position" rule as a single fact — it resolves every reverse-alphabet question instantly, without having to count backward from Z each time.

Worked example — number coding. If in a certain code CHAIR is written as 3-8-1-9-18, how would TABLE be written?

Fast method: The code is simply each letter's position in the alphabet (A=1, B=2, and so on): C=3, H=8, A=1, I=9, R=18 — matches exactly. Apply the same rule to TABLE: T=20, A=1, B=2, L=12, E=5. Code = 20-1-2-12-5. This sub-type needs no shift at all, only the direct alphabet-position table, which is worth memorising in blocks of five (A-E = 1-5, F-J = 6-10, K-O = 11-15, P-T = 16-20, U-Z = 21-26) so you can recall any letter's number within a second or two.

4. Blood Relations — the Family-Tree Sketch Method

Blood relation questions are lost not to reasoning error but to holding too much information in your head. The fix is mechanical: draw a small family tree using symbols the instant you start reading the question, and update it as each new relation is introduced.

Worked example. Pointing to a photograph, a man said, "She is the daughter of my grandfather's only son." How is the woman related to the man?

Fast method: "My grandfather's only son" — since the man himself is a descendant of this grandfather, and the son is described as "only," the son must be the man's father (the man's father is his grandfather's son; if the son were someone else, the man wouldn't logically be in the family line through this grandfather in the standard reading of these questions). So "daughter of my father" is either the man's sister or the man himself if female — since "she" is specified, it is the man's sister. Draw it as: Grandfather → Father (only son) → [Man, She]. The answer is Sister. The sketch prevents the common error of assuming "grandfather's son" could be an uncle — the word "only" is the key constraint that forces the son to be the man's own father.

Worked example — using symbols. A + B means A is the mother of B; A - B means A is the brother of B; A × B means A is the wife of B. If P + Q - R, how is P related to R?

Fast method: P + Q means P is mother of Q. Q - R means Q is brother of R. So P is the mother of Q, and Q is R's brother, which makes P the mother of R as well (since Q and R share the same mother, being siblings via the brother relation). Answer: P is the mother of R. Writing the symbolic chain left to right and translating each symbol in sequence, without trying to hold the full sentence in memory, is what keeps this fast.

5. Direction Sense — the Coordinate-Grid Shortcut

Direction questions ask you to track a person's position after a series of turns and movements. Do not visualise this in your head — plot it on an imaginary (or literally sketched, in the margin) coordinate grid where East is positive x, North is positive y.

Worked example. Raju walks 5 km North, then turns right and walks 3 km, then turns right again and walks 5 km. How far is he from his starting point, and in which direction?

Fast method: Start at (0,0). Walk 5 km North: now at (0,5). Facing North, turn right means now facing East; walk 3 km East: now at (3,5). Facing East, turn right again means now facing South; walk 5 km South: now at (3,0). Straight-line distance from start (0,0) to (3,0) is simply 3 km, and the direction from start to current position is due East. The key insight: the two 5 km legs (North then South) cancel out entirely in the y-coordinate, leaving only the 3 km East leg as net displacement — you do not need Pythagoras here because the y-displacement is zero.

Worked example — with Pythagoras needed. A man walks 8 km East and then 6 km North. How far is he from the starting point?

Fast method: Position is (8,6) relative to start (0,0). Distance = square root of (8² + 6²) = square root of (64+36) = square root of 100 = 10 km. Recognising the 6-8-10 Pythagorean triple instantly (a scaled version of 3-4-5) avoids having to compute a square root from scratch — memorise the common triples: 3-4-5, 6-8-10, 5-12-13, 9-12-15, 8-15-17.

6. Syllogism — the Venn Diagram Elimination Method

Syllogism is the sub-type most vulnerable to "sounds right" errors, because a conclusion can feel intuitively true in everyday language while being logically invalid given only the stated premises. The fix is to draw a Venn diagram for every possible arrangement the premises allow, not just the one that first comes to mind.

Worked example. Statements: All cats are dogs. All dogs are animals. Conclusions: I. All cats are animals. II. Some animals are cats.

Fast method: Draw three nested circles: Cats inside Dogs inside Animals (since "all cats are dogs" nests cats fully within dogs, and "all dogs are animals" nests dogs fully within animals). From this single valid diagram, "all cats are animals" is directly visible (Conclusion I follows). "Some animals are cats" is also true, because the cats circle, being non-empty and inside animals, guarantees an overlap (Conclusion II also follows). Both conclusions follow. The nested-circle diagram, drawn once, answers both conclusions simultaneously — there is no need to test them one at a time with separate logical chains.

Worked example — a case with no valid conclusion. Statements: Some pens are pencils. Some pencils are erasers. Conclusion: Some pens are erasers.

Fast method: "Some pens are pencils" allows a diagram where the pen-pencil overlap and the pencil-eraser overlap are in completely different parts of the pencils circle, with no forced overlap between pens and erasers. Since at least one valid diagram exists where pens and erasers do not overlap at all, the conclusion "some pens are erasers" does not follow. The rule of thumb: a conclusion is valid only if it holds in every possible diagram consistent with the premises, not just in the most "natural-looking" one — always actively try to draw a diagram that breaks the conclusion before accepting it.

7. Seating Arrangement — the Elimination-Grid Method

Seating arrangement is the most time-expensive reasoning sub-type if attempted linearly (reading clue by clue and redrawing each time). The fast method is to build a grid of all seats and candidates, then eliminate cells clue by clue, rather than trying to place people directly.

Worked example. Five friends A, B, C, D, E sit in a row facing North. B sits second from the left end. D sits immediately to the right of B. A sits at one of the extreme ends. C sits immediately left of E. Find the seating order.

Fast method: Fix the five positions 1 to 5 left to right. B is at position 2 (fixed directly by the clue — always place absolute-position clues on the grid first, before relative clues). D is immediately right of B, so D is at position 3. A is at an extreme end, so A is at position 1 or position 5; since position 2 and 3 are taken by B and D, and C must be immediately left of E (a pair that needs two adjacent open seats), check which extreme leaves room for the C-E pair. If A is at position 1, remaining seats 4 and 5 must hold C and E with C immediately left of E, giving C at 4 and E at 5 — this works cleanly. If A were at position 5 instead, the only remaining adjacent pair would be positions 4 and... position 5 is now taken by A, so C-E cannot both fit adjacently in the remaining seats (1 is isolated from 4). So A must be at position 1. Final order: A, B, D, C, E. The discipline of placing every absolute clue on the grid before touching relative clues is what prevents having to backtrack and redraw.

8. Non-Verbal Pattern Questions — the Rotation and Count Shortcut

Figure series and mirror/water-image questions reward two specific habits: counting elements (dots, lines, sides) rather than trying to "see" the whole pattern at once, and knowing the fixed rules for mirror and water images without re-deriving them each time.

Mirror image rule: A vertical mirror (held to the right or left of a figure) flips the figure left-right, keeping top and bottom unchanged. A horizontal mirror (held above or below) flips top-bottom, keeping left-right unchanged.

Water image rule: A water image is always a top-bottom (vertical axis of reflection is horizontal) flip, as though the figure is reflected in a pool below it — equivalent to a horizontal mirror placed at the bottom of the figure.

Worked example — element counting. In a figure series, each successive figure has one more dot than the last, and the dots are arranged in a rotating position around a fixed square. If figure 1 has 2 dots and figure 4 has 5 dots, how many dots will figure 6 have?

Fast method: The dot count increases by exactly 1 per figure (confirmed by figure 1 → figure 4 going from 2 to 5, a gain of 3 over 3 steps, i.e. 1 per step). Figure 6 is 2 steps beyond figure 4, so dots = 5+2 = 7. Counting the numeric attribute (dots) and tracking it as a simple series is far faster than trying to visually track rotational position and count simultaneously.

9. Elimination-First Strategy for All Reasoning MCQs

Across every reasoning sub-type, the fastest solvers share one habit: they scan the four answer options before finalising a full solution, because reasoning questions are frequently solvable by testing options against the given conditions rather than deriving the answer from first principles.

Worked example. If in a certain code language, "PEN" is written as "RGP," what is the code for "BOX"?
(a) DQZ (b) DQY (c) CQZ (d) DPZ

Fast method: Rather than deriving the rule from scratch, notice P→R is +2, E→G is +2, N→P is +2 — a consistent +2 shift confirmed across all three letters of the sample word. Apply +2 to BOX: B→D, O→Q, X→Z, giving DQZ, which is option (a). Once you have the rule from the worked pair, checking it against the answer options (rather than fully re-deriving from the alphabet each letter) is often the fastest path, especially when options differ by only one letter, because a single correct letter placement is often enough to identify the right option and eliminate the rest.

10. Ranking and Ordering — the Number-Line Method

Questions that ask "in a row of children, X is 7th from the left and 12th from the right; how many children are in the row" are pure arithmetic dressed up as reasoning, and the fast method treats the row as a number line rather than trying to visualise children standing in a line.

Worked example. In a row of students, Ravi is 7th from the left end and 12th from the right end. How many students are in the row?

Fast method: Total = (position from left) + (position from right) - 1, because Ravi himself is counted once in each count and must be subtracted from the sum. Total = 7 + 12 - 1 = 18 students. The "-1" is the single detail every candidate forgets under pressure, and it is worth over-learning as a fixed part of the formula rather than re-deriving it each time.

Worked example — two people, ranks from opposite ends. In a class of 40 students, Meena ranks 15th from the top. What is her rank from the bottom?

Fast method: Rank from bottom = Total - Rank from top + 1 = 40 - 15 + 1 = 26. Again, the "+1" corrects for double-counting Meena's own position. Committing this formula to memory in exactly this form avoids the classic off-by-one error that costs marks on an otherwise trivial question.

11. Analogy — the Relationship-First Method

Analogy questions ask you to complete "A is to B as C is to ?" and the fast method is to state the relationship between A and B in words before looking at the answer options, rather than scanning the options first and trying to reverse-engineer a fit.

Worked example. Doctor is to Hospital as Teacher is to ?
(a) Student (b) School (c) Blackboard (d) Book

Fast method: State the relationship explicitly first: "A doctor works at a hospital." Applying the identical relationship: "A teacher works at a ___." The answer must be the workplace, which is School, option (b). Candidates who skip the explicit verbal statement and jump straight to scanning options often get pulled toward "Student" because it is the most strongly associated word with "Teacher," even though the relationship being tested is workplace, not the people served.

Worked example — numeric analogy. 8 is to 64 as 9 is to ?
(a) 72 (b) 81 (c) 90 (d) 99

Fast method: Relationship: 64 = 8², a squaring relationship. Applying to 9: 9² = 81, option (b). Numeric analogies almost always test one of a small set of relationships — squaring, cubing, doubling, or a fixed addition/multiplication — so check these in order rather than searching randomly.

12. Classification (Odd One Out) — the Category-Naming Method

Odd-one-out questions are fastest when you name the shared category of the majority explicitly, rather than comparing each option to every other option pairwise.

Worked example. Find the odd one out: Delhi, Mumbai, Chennai, Sikkim.
(a) Delhi (b) Mumbai (c) Chennai (d) Sikkim

Fast method: Name the category of three that clearly match: Delhi, Mumbai, and Chennai are all cities. Sikkim is a state, not a city. Answer: (d) Sikkim. Once you can name why three items belong together, the fourth stands out immediately — this is faster than checking six possible pairs for similarity.

13. Calendar Basics — the Odd-Days Shortcut

"What day of the week was/will be a given date" questions rely on counting odd days, and the shortcut is a small set of memorised facts rather than counting every day manually.

Key facts to memorise: An ordinary year has 365 days = 52 weeks + 1 odd day. A leap year has 366 days = 52 weeks + 2 odd days. 100 years have 5 odd days, 200 years have 3 odd days, 300 years have 1 odd day, and 400 years have 0 odd days (400 years is an exact multiple of weeks).

Worked example. If 1 January 2024 was a Monday, what day was 1 January 2025?

Fast method: 2024 is a leap year (divisible by 4, and not a century year, so no further check needed), so it contributes 2 odd days. Starting from Monday, add 2 odd days: Monday+1=Tuesday, Tuesday+1=Wednesday. 1 January 2025 was a Wednesday. The entire calculation reduces to identifying whether the elapsed year was a leap year and adding the corresponding 1 or 2 odd days to the known starting day — no need to count 366 individual days.

14. Building Reasoning Speed: A Practical Drill Routine

Reasoning speed comes from pattern-recognition automation, and the fastest way to build it is repeated exposure to the same structural types under a timer, rather than reading more theory. Three drills structure this well in the final weeks before the exam.

Drill one — the ninety-second series sprint. Solve ten mixed number and letter series in ninety seconds total. This forces you to apply the difference-of-differences check as a reflex rather than a deliberate step, because there is no time to consciously decide which method to try.

Drill two — the diagram-first syllogism drill. Take ten syllogism statement-conclusion sets and, for each, force yourself to sketch the Venn diagram before reading the conclusions. This breaks the habit of reasoning verbally about syllogisms, which is the single biggest source of "sounds right but is logically invalid" errors on this sub-type.

Drill three — the seating-arrangement grid drill. Practice five seating arrangement puzzles using strictly the grid-and-eliminate method described above, timing how long it takes to go from reading the clues to filling every seat. Candidates who make this method automatic typically cut seating-arrangement time by more than half compared to trial-and-error placement.

As with the arithmetic chapter, none of this requires new theory once you have read the techniques above — it requires enough repetition that recognising "this is a difference-of-differences series" or "this needs a nested-circle diagram" happens in the first two seconds of reading the question, leaving the remaining time purely for mechanical execution.

Speed-Technique Checklist

• For number series, always compute the difference-of-differences before trying ratios or any other operation.
• For letter series, convert every letter to its alphabet position number before looking for a pattern.
• For coding-decoding, confirm the shift or rule using just two or three letter pairs, then apply it directly — do not re-derive for every letter.
• Memorise the "27 minus position" rule for reverse-alphabet coding.
• For blood relations, sketch a family tree with symbols the moment new information appears — never hold more than one relation in your head at a time.
• For direction questions, plot movements on an imaginary coordinate grid; check whether displacement legs cancel before reaching for Pythagoras.
• Memorise Pythagorean triples 3-4-5, 6-8-10, 5-12-13, 9-12-15, 8-15-17 to skip square-root computation.
• For syllogism, always attempt to draw a diagram that breaks a conclusion before accepting it as valid.
• For seating arrangement, place all absolute-position clues on the grid first, relative clues second.
• For non-verbal series, count a single numeric attribute (dots, sides, lines) rather than tracking the whole figure at once.
• Scan the answer options before committing to a full derivation — reasoning MCQs are often faster to solve by option-testing.

Practice MCQs

1. Find the next term: 4, 9, 16, 25, 36, ?
   • (a) 42
   • (b) 45
   • (c) 49
   • (d) 54
   Answer: (c) 49. Explanation: The series is squares of 2,3,4,5,6; next is 7² = 49.
2. Find the next term: 7, 12, 20, 31, 45, ?
   • (a) 58
   • (b) 60
   • (c) 62
   • (d) 64
   Answer: (c) 62. Explanation: Differences 5,8,11,14 increase by 3; next difference 17, 45+17=62.
3. Find the next letter: C, F, J, O, ?
   • (a) T
   • (b) S
   • (c) U
   • (d) R
   Answer: (c) U. Explanation: Positions 3,6,10,15 have differences 3,4,5; the next difference is 6, giving 15+6 = 21, which is the letter U.
4. If in a code, DELHI is written as FGNJK, what is the code for MUMBAI?
   • (a) OWOCDK
   • (b) OWODCK
   • (c) OWNDCK
   • (d) NWODCK
   Answer: (b) OWODCK. Explanation: Each letter shifts forward by 2 (confirmed by D→F, E→G, L→N in the sample word). Applying the same shift: M→O, U→W, M→O, B→D, A→C, I→K, giving OWODCK.
5. Pointing to a man, a woman said, "His mother is the only daughter of my mother." How is the woman related to the man?
   • (a) Mother
   • (b) Sister
   • (c) Aunt
   • (d) Grandmother
   Answer: (a) Mother. Explanation: "Only daughter of my mother" is the woman herself, so the man's mother is the woman.
6. A man walks 6 km East and then 8 km North. How far is he from the starting point?
   • (a) 8 km
   • (b) 9 km
   • (c) 10 km
   • (d) 12 km
   Answer: (c) 10 km. Explanation: Pythagorean triple 6-8-10; square root of (36+64) = square root of 100 = 10.
7. Statements: All roses are flowers. Some flowers are red. Conclusion: Some roses are red.
   • (a) Conclusion follows
   • (b) Conclusion does not follow
   • (c) Cannot be determined
   • (d) Both follow and do not follow
   Answer: (b) Conclusion does not follow. Explanation: The red flowers and roses may occupy non-overlapping portions of the flowers circle; not every diagram forces an overlap.
8. Five people sit in a row. R sits at the extreme left. S sits immediately right of R. T sits immediately right of S. If U sits immediately right of T and V is at the extreme right, what is the order?
   • (a) R,S,T,U,V
   • (b) R,T,S,U,V
   • (c) S,R,T,U,V
   • (d) R,S,U,T,V
   Answer: (a) R,S,T,U,V. Explanation: Each clue fixes the next seat directly to the right in sequence.
9. In a mirror held vertically to the right of a figure, what changes?
   • (a) Top and bottom swap
   • (b) Left and right swap
   • (c) Nothing changes
   • (d) The figure rotates 180 degrees
   Answer: (b) Left and right swap. Explanation: A vertical mirror flips left-right, keeping top-bottom unchanged.
10. If GOLD is coded using the reverse-alphabet rule (A=Z, B=Y, ...), what is the code?
    • (a) TLOW
    • (b) TLOV
    • (c) SLOW
    • (d) TLPW
    Answer: (a) TLOW. Explanation: Using code position = 27 - original position: G(7)→T(20), O(15)→L(12), L(12)→O(15), D(4)→W(23).
11. A + B means A is mother of B; A - B means A is brother of B. If X + Y - Z, how is X related to Z?
    • (a) Aunt
    • (b) Sister
    • (c) Mother
    • (d) Grandmother
    Answer: (c) Mother. Explanation: X is mother of Y, and Y is brother of Z, so X is also Z's mother.
12. Find the odd one out: 8, 27, 64, 100, 125.
    • (a) 27
    • (b) 64
    • (c) 100
    • (d) 125
    Answer: (c) 100. Explanation: 8=2³, 27=3³, 64=4³, 125=5³ are perfect cubes; 100 is not.
13. Raju walks 4 km South, turns left, walks 4 km, turns left again, walks 4 km. How far is he from the start?
    • (a) 0 km
    • (b) 4 km
    • (c) 8 km
    • (d) 12 km
    Answer: (b) 4 km. Explanation: South then East then North for 4 km each cancels the North-South leg, leaving 4 km East displacement.
14. Statements: All pens are books. All books are shelves. Conclusion I: All pens are shelves. Conclusion II: Some shelves are pens.
    • (a) Only I follows
    • (b) Only II follows
    • (c) Both follow
    • (d) Neither follows
    Answer: (c) Both follow. Explanation: Nested circles pens-inside-books-inside-shelves make both conclusions directly visible.
15. If PEN is coded as RGP, what is BOX coded as?
    • (a) DQZ
    • (b) DQY
    • (c) CQZ
    • (d) DPZ
    Answer: (a) DQZ. Explanation: The shift is +2 for each letter; B→D, O→Q, X→Z.