CAT DILR Practice Tool — All 20 Topics.

TRICK: For any sequence, check differences first. If differences are constant → AP. If differences double → 2ⁿ pattern. If second differences are constant → quadratic.

TRICK: Perfect squares series — 1,4,9,16,25... differences are always consecutive odd numbers: 3,5,7,9,11.

TRICK: Fibonacci — if first differences equal previous terms, it's Fibonacci or a variant.

TRICK: For ×2+1, ×2+1 pattern: 1,3,7,15,31,63... each term = 2ⁿ−1.

TRICK: n(n+1) series — 2,6,12,20,30,42... differences are 4,6,8,10,12 (even, increasing by 2).

TRICK: Triangular numbers — T(n)=n(n+1)/2: 1,3,6,10,15,21,28... each difference increases by 1.

TRICK: Cube pattern — 1,8,27,64,125: differences 7,19,37,61 (second differences 12,18,24 — constant third difference).

TRICK: Alternating series — two interleaved series. Separate odd-positioned and even-positioned terms and find each pattern independently.

TRICK: 'Only son/daughter of my parent' = ME. This eliminates all ambiguity in complex relation chains.

TRICK: Always draw a family tree. Male = square, Female = circle. One generation per horizontal level.

TRICK: Coded relations — decode each symbol step by step. Never jump to conclusion before mapping all symbols.

TRICK: 'My father's wife' = my mother (unless stepfamily mentioned). 'My mother's husband' = my father.

TRICK: Brother/sister of father = uncle/aunt. Brother/sister of mother = maternal uncle/aunt. Their children = cousins.

TRICK: For 'pointing to a photo' problems: use the actual statement to trace one step at a time. Don't assume gender unless explicitly stated.

TRICK: If the chain goes X→parent→child→X again, it's a loop — re-read the problem. Circular references are always traps.

TRICK: Generation counting — every parent-child link goes one generation. Cousin = same generation (0 steps), uncle/nephew = 1 generation gap.

TRICK: Hour hand moves 0.5°/min. Minute hand moves 6°/min. Relative speed of minute hand over hour hand = 5.5°/min.

TRICK: Mirror time formula — if clock shows T, mirror shows (11:60 − T). So 3:15 mirror = 8:45.

TRICK: Hands coincide every 65 5/11 minutes (not every 60 minutes). In 12 hours: 11 coincidences.

TRICK: Hands at 90° — this happens 22 times in 12 hours (not 24). Happens at slightly irregular intervals.

TRICK: Odd days — Jan=3, Feb=0(ordinary)/1(leap), Mar=3, Apr=2, May=3, Jun=2, Jul=3, Aug=3, Sep=2, Oct=3, Nov=2, Dec=3.

TRICK: Day of week — use anchor dates. Jan 1, 2000 = Saturday. Add odd days to find any date.

TRICK: Century odd days — 100 years = 5 odd days. 200 years = 3. 300 years = 1. 400 years = 0.

TRICK: Fast/slow clocks — if a clock gains X min per hour, in T hours it gains X×T minutes. For 'true time', subtract gained time.

TRICK: Categories to check — (1) prime vs composite, (2) perfect squares/cubes, (3) odd vs even, (4) multiples of a number, (5) alphabetical property.

TRICK: For word groups — check (1) synonyms, (2) category membership, (3) number of letters, (4) vowels count, (5) anagram relationships.

TRICK: Number groups — convert all to the same form (all as squares, all as primes, all as ratios) before comparing.

TRICK: Letter groups — write out alphabetical positions. Check if positions form a pattern. The odd one breaks the pattern.

TRICK: If the group seems to have 2 odd ones, re-examine the pattern. In well-set questions, exactly 1 does not fit.

TRICK: For scientific/general knowledge odd-one-out, group by: (1) classification, (2) property, (3) usage. The answer is always a different CLASS, not just a different example.

TRICK: Identify the shift immediately — compare input letters to output. Check if shift is constant (+N or −N).

TRICK: Reverse alphabet codes — A=Z, B=Y... use formula: coded position = 27 − original position.

TRICK: If word is reversed first, decode by reversing the coded word first, then apply the letter shift.

TRICK: Number codes — if numbers represent letter positions, 1=A, 2=B... check if the number directly = alphabetical position.

TRICK: In multiplication/operator swap codes, identify all operations first. Map each coded operation to its real meaning before computing.

TRICK: ROT13 is self-inverting — applying ROT13 twice gives original. Shift of 13 is special because 26/2=13.

TRICK: For matrix/grid codes — find the row-column pattern. Each letter's position in the grid determines its code.

TRICK: Positional shift codes — where letter N in the word gets shifted by N — decode by subtracting position from coded letter.

TRICK: Vertical mirror — left becomes right, right becomes left. Top-bottom stays same. Think: FOLD the paper along a vertical line.

TRICK: Water image (horizontal mirror) — top becomes bottom, bottom becomes top. Left-right stays same.

TRICK: Clock mirror formula: Mirror time = 11:60 − Original time. Example: 4:25 → 7:35.

TRICK: Symmetric letters (vertical): A H I M O T U V W X Y — these look identical in a vertical mirror.

TRICK: Symmetric letters (horizontal): B C D E H I K O X — these look identical in a water image.

TRICK: Two reflections in the same axis = back to original. Vertical mirror + vertical mirror = original.

TRICK: Vertical mirror + horizontal mirror = 180° rotation. This is a useful shortcut for complex reflection questions.

TRICK: Transparent sheet held to mirror — the text appears CORRECT (double reversal cancels out). The sheet itself shows a mirror image.

TRICK: Clockwise rotation directions — N→E→S→W→N. Anti-clockwise — N→W→S→E→N.

TRICK: 90° CW = same as 270° ACW. 270° CW = same as 90° ACW. Always convert to the smaller equivalent.

TRICK: Two reflections = rotation. Vertical mirror + Horizontal mirror = 180° rotation.

TRICK: Letter rotations — N rotated 180° = N. S rotated 180° = S. Z rotated 90° CW = N.

TRICK: For figures with n-fold symmetry, minimum rotation = 360°/n. Square: 90°. Equilateral triangle: 120°. Regular hexagon: 60°.

TRICK: Dice roll directions — rolling forward: top→front, front→bottom, bottom→back, back→top. Right column unchanged.

TRICK: Clock rotated 180° — each number N maps to its position diametrically opposite. 3 goes to 9 position, reads 9:00.

TRICK: Net rotation = sum of all CW rotations − sum of all ACW rotations. Simplify using mod 360.

TRICK: All A are B + All B are C → All A are C. (Universal chain — strongest conclusion.)

TRICK: Some A are B + All B are C → Some A are C. (Some transfers through universal.)

TRICK: All A are B + No B is C → No A is C. (Universal negative blocks everything.)

TRICK: Some A are B + No B is C → Some A are NOT C. (But cannot say ALL A are not C.)

TRICK: 'Some A are B' is always convertible to 'Some B are A'. Never assume the reverse for All or No.

TRICK: For 'Either conclusion I or II follows' — if one is the exact negation of the other, one of them must follow. Always check this.

TRICK: Never add extra information. 'All cats are mammals' does NOT mean 'All mammals are cats'.

TRICK: Complementary pairs — 'Some A are B' and 'No A is B' are contradictory. If neither follows separately, check if they form a complementary pair.

TRICK: Use a coordinate grid. North = +Y, South = −Y, East = +X, West = −X. Displacement = √(X² + Y²).

TRICK: Right turn from N→E→S→W. Left turn from N→W→S→E. Memorise this as clockwise/anti-clockwise.

TRICK: If total East = total West, the person is on the same E-W line as start. Similarly for North-South.

TRICK: Pythagorean triples frequently appear — 3-4-5, 5-12-13, 8-15-17. Recognise these for quick distance calculation.

TRICK: Final direction — only track the LAST direction faced, not the displacement. These are two different questions.

TRICK: Facing each other — if two people face each other, their directions are exactly opposite.

TRICK: Shadow direction — morning sun is in the East, so shadows fall West. Evening sun in West, shadows fall East. At noon, shadow falls South (in Northern hemisphere).

TRICK: After n complete right turns = n×90° rotation = back to original direction if n=4. After 3 right turns = same as 1 left turn.

TRICK: Key formula — n(A∪B) = n(A) + n(B) − n(A∩B). Rearrange to find any missing value.

TRICK: Neither = Total − n(A∪B). Always calculate 'at least one' first, then subtract from total.

TRICK: For three sets: n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C).

TRICK: 'Only A' = n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C). Subtract all overlaps, add back triple-overlap.

TRICK: If n(A∪B) = n(A) + n(B), the sets are MUTUALLY EXCLUSIVE (no overlap).

TRICK: If n(A∩B) = n(A), then A is a SUBSET of B (A is completely inside B).

TRICK: Percentage problems — treat total = 100%. Apply all formulas the same way with percentages.

TRICK: For probability form — P(A∪B) = P(A)+P(B)−P(A∩B). Mutually exclusive: P(A∪B) = P(A)+P(B).

TRICK: First identify the RELATIONSHIP type: tool-use, part-whole, category-member, cause-effect, synonym, antonym, worker-workplace, young-one, scientific-study.

TRICK: Tool:Use — Pen:Write, Knife:Cut, Needle:Sew, Scissors:Cut, Chisel:Carve.

TRICK: Worker:Workplace — Doctor:Hospital, Teacher:School, Lawyer:Court, Soldier:Barracks, Chef:Kitchen.

TRICK: Study:Subject — Seismology:Earthquakes, Cardiology:Heart, Ornithology:Birds, Numismatics:Coins, Philately:Stamps.

TRICK: Young ones — Cow:Calf, Horse:Foal, Dog:Puppy, Cat:Kitten, Lion:Cub, Duck:Duckling, Sheep:Lamb.

TRICK: Containers — Sword:Scabbard, Gun:Holster, Arrow:Quiver, Wine:Bottle, Grain:Silo.

TRICK: For vocabulary analogies — lachrymose=teary, lugubrious=mournful, misanthrope=hates people, philanderer=womaniser, bibliophile=book lover.

TRICK: Letter analogies — find the numerical position of each letter. The relationship is always arithmetic (add, subtract, multiply positions).

TRICK: AND = multiplication (1×1=1, anything ×0=0). OR = addition with cap at 1 (0+0=0, otherwise 1).

TRICK: XOR = 'same gives 0, different gives 1'. Shortcut: XOR is addition mod 2.

TRICK: De Morgan's Laws — NOT(A AND B) = NOT A OR NOT B. NOT(A OR B) = NOT A AND NOT B.

TRICK: A AND (NOT A) = always 0 (contradiction). A OR (NOT A) = always 1 (tautology).

TRICK: NAND and NOR are 'universal gates' — any logic circuit can be built using only NAND gates or only NOR gates.

TRICK: Double negation — NOT(NOT A) = A. Always simplify double negations first.

TRICK: Operator precedence — NOT first, then AND, then OR. Brackets override all.

TRICK: For Knights-and-Knaves logic — if a statement creates a contradiction regardless of who says it, the speaker is the Knave.

TRICK: For an n×n×n painted cube cut into unit cubes: • 3 faces painted (corners): always 8 • 2 faces painted (edges): 12(n−2) • 1 face painted (face centres): 6(n−2)² • 0 faces painted (interior): (n−2)³

TRICK: Total unit cubes = n³. Cross-check: 8 + 12(n−2) + 6(n−2)² + (n−2)³ = n³. Always verify.

TRICK: Opposite faces on standard dice sum to 7: 1+6=7, 2+5=7, 3+4=7. This is the most-tested dice fact.

TRICK: Dice rolling — rolling FORWARD: top→front, front→bottom, bottom→back, back→top. Left and right stay.

TRICK: A cube has exactly 11 distinct nets. You don't need to memorise all 11 — just know which shapes CANNOT fold (Z-shape staircases, 2×3 blocks with awkward arrangement).

TRICK: For 3×3×3 cube painted in split colours (3 faces one colour, 3 another): corner pieces at the junction of both colour sets have BOTH colours painted.

TRICK: Probability from painted cube = (number of pieces with condition) / n³. For corners: always 8/n³.

TRICK: For 'how many have at least 1 face painted' = n³ − (n−2)³ = all painted pieces.

TRICK: Position from other end = (Total + 1) − Position from this end. Always add 1.

TRICK: Total in row = Position from left + Position from right − 1. (Because the person is counted twice.)

TRICK: Between two people: people between = |Position A − Position B| − 1.

TRICK: If 'n people join in front', add n to position from front. If 'n people join at back', add n to position from back.

TRICK: Ranking chains — convert all comparisons to a single ordered chain. A>B>C>D means A=1st, D=last.

TRICK: If rank from top + rank from bottom = n+1, total people = n. Classic identity used in reverse.

TRICK: For 'n people removed from front', subtract n from front-position. Ensure it doesn't go below 1.

TRICK: Circular arrangements — rank from left is undefined. Questions will specify 'to the right of' or 'to the left of' a reference person.

TRICK: Multiplying or dividing both sides by a NEGATIVE number reverses the inequality. This is the most common trap.

TRICK: If a > b and both positive, then 1/a < 1/b. The reciprocal reverses the inequality for positives.

TRICK: For coded inequality questions, decode ALL symbols first. Make a clean list: @=≥, #=≤, etc. Then solve.

TRICK: Transitive property — if A≥B and B>C, then A>C. The STRONGER of the two relationships is the conclusion.

TRICK: (a+1)/(b+1) < a/b when a>b>0. Cross-multiply to verify: b(a+1) < a(b+1) iff b < a.

TRICK: AM-GM inequality — for positive x: x + 1/x ≥ 2, with equality when x=1.

TRICK: |x| < k means −k < x < k. |x| > k means x > k OR x < −k. These two forms cover all absolute value inequality questions.

TRICK: For compound statements like 'P@Q, Q#R' — draw the inequality chain visually. Only conclude what the chain directly supports.

TRICK: First separate letters from numbers. Find the letter pattern independently, then the number pattern independently.

TRICK: Letter positions A=1, B=2, C=3... Z=26. Most patterns use: +1, +2, skip-one, or reverse-alphabetical positions.

TRICK: If letters skip one: A,C,E,G,I (odd positions) or B,D,F,H (even positions). If letters skip two: A,D,G,J (positions 1,4,7,10).

TRICK: Numbers in alphanumeric series often follow: perfect squares (1,4,9,16,25), triangular numbers (1,3,6,10,15), or simple AP/GP.

TRICK: For pairs like AZ, BY, CX — letters from front and back of alphabet pairing up. Positions sum to 27 (A+Z=1+26=27).

TRICK: Pattern with reverse pairs: A1Z26, C3X24 — first letter advances, last letter retreats, numbers track letter positions.

TRICK: In double-letter series like AB12, CD34, EF56 — letters advance in consecutive pairs, numbers advance in consecutive digit pairs.

TRICK: Always write out the full decoded pattern before selecting the answer. Never rush at the last term.

TRICK: Read ALL conditions before evaluating. Check each condition independently, then combine.

TRICK: Look for exceptions/relaxation clauses AFTER checking if the candidate fails any condition. Many candidates qualify via exceptions.

TRICK: Note words like 'minimum', 'maximum', 'at least', 'not more than' — these define the boundary exactly. 60% minimum means 60% is acceptable.

TRICK: AND conditions — ALL must be satisfied. OR conditions — at least ONE must be satisfied.

TRICK: Exception hierarchy — if a special rule applies (SC/ST, ex-serviceman, PhD, etc.), check if the candidate qualifies for that exception FIRST.

TRICK: Elimination order — start with the easiest/most restrictive condition. If it fails with no exception, immediately mark as 'not eligible'.

TRICK: For weighted scoring — calculate the total score, compare against the stated minimum. Also check individual minimums separately.

TRICK: 'Referred to committee/review board' is a THIRD outcome besides selected/rejected. Watch for this in complex selection problems.

TRICK: n folds = 2ⁿ layers. Layers determine the number of holes: 1 punch through 2ⁿ layers = 2ⁿ holes.

TRICK: Right-over-left fold, then top-over-bottom fold = all 4 quadrants concentrated in one small square. Any punch = 4 holes symmetrically placed.

TRICK: Punch at a CORNER of a twice-folded paper = 4 holes, one at each corner of the original paper.

TRICK: Punch at the CENTRE of a twice-folded paper = 4 holes, one in each quadrant of the original.

TRICK: Accordion fold (3 layers) then one regular fold (×2) = 6 layers total. Not 3+2=5. It multiplies.

TRICK: Cutting along the FOLD LINE separates the folded halves. Cutting PARALLEL to fold lines creates symmetric notches.

TRICK: Semicircle cut from a folded edge = full circle when unfolded. Triangle cut from corner = two symmetric triangular notches.

TRICK: Number of pieces = number of cuts through all layers + 1 (for each separate cut line).

TRICK: Check 3 things in order: (1) row patterns, (2) column patterns, (3) diagonal patterns. The answer must satisfy at least one consistently.

TRICK: For number matrices — check row sums, row products, column sums. Often one operation is constant across all rows/columns.

TRICK: For shape matrices — identify the RULE: rotation, reflection, size change, shading change, or addition/removal of elements.

TRICK: Latin Square rule — each row and column has each element exactly once. Use elimination: if a shape appears in the row AND column, it cannot be the answer.

TRICK: Progressive shading — count shaded cells per row and per column. The missing cell's shading count must complete the pattern.

TRICK: Size doubling in matrix — if shapes double in size row by row, check both dimensions. Area quadruples when linear size doubles.

TRICK: For R(i,j) = f(i,j) formulas — identify f by computing a few known cells. Then apply to find missing cell.

TRICK: Anti-diagonal patterns are as important as main diagonal. Check both: top-right to bottom-left is often the key relationship.

TRICK: First understand the OPERATION, then apply it step by step. Never guess from one example — always verify with 2+ examples.

TRICK: Common operations: sort ascending/descending, reverse string, shift letters, apply arithmetic, swap adjacent elements.

TRICK: For multi-step machines — apply steps in ORDER. Keep intermediate results clean.

TRICK: If input→output: check differences (arithmetic), ratios (geometric), letter shifts, or positional rearrangements.

TRICK: For pattern-based output rules (input 4→18, 7→51, 3→11) — test n², n³, n²+k, n³+k systematically.

TRICK: Fibonacci machine — if each output depends on previous two, write the sequence out fully from the given starting values.

TRICK: String operations — reversal, capitalisation, vowel removal/replacement are the three most common. Check each operation's scope (whole word vs each word).

TRICK: Even/Odd machine — applying an even/odd transformation twice often returns to the original. Look for cycles of length 2.