Cheatsheet

• % more/less than B = A − B/B × 100 — base = value after than
• Pie: value = Total × %/100 or Total × angle/360; same-pie ratio = ratio of %s
• % change = New − Old/Old × 100; Sum = Average × Count
• Ratio + one part: M : F = 7 : 5, F = 25 → 1 unit = 5 → M = 35, total = 60
• Two pies (together vs one part): other = together − that part, sector by sector
• Total = Part 1 + Part 2 (males + females, sold + unsold) → find the missing part

FastKeep fractions (75% = 3/4, 45% = 9/20, 37.5% = 3/8); write only the values each question needs

TrapWrong base or units ('00, thousand, lakh); 150% of B is only 50% more than B; total taken instead of a part

• Slow growth → differences; fast growth (≈ ×2 or more) → multiply; up-down → alternate
• Differences look random? Take 2nd differences; check squares, cubes, primes, n² ± 1
• Fast growth: divide each term by the one before → ×n; leftover (term − prev × n) = add-on
• Decimals (2.5, 4.5, 12…) usually mean ×0.5, ×1.5, ×2.5 type multipliers
• Wrong number: fit a rule on 4–5 terms from one end, rebuild; first mismatch = answer
• Gap in the middle: test each option against both neighbours

FastWrite the differences under the terms; in big × series the unit digit often settles the option

TrapCheck the rule on every term incl. the last; only one term is wrong — fix the one that fixes both odd diffs

• BODMAS: of before ÷ and ×: 140 ÷ 4 of 7 = 140 ÷ 28 = 5
• ÷ and × at the same level → left to right: 32 ÷ 4 × 128 = 8 × 128 = 1024
• Approximation: round every number to nearest whole first (1499.98 → 1500, 7.01 → 7)
• x% of y = y% of x (18% of 650 = 650% of 18); a² − b² = (a − b)(a + b)
• aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; 2⁷ = 128, 2¹⁰ = 1024
• Squares to 40² (38² = 1444), cubes to 15³ (12³ = 1728, 14³ = 2744)

FastSolve the known side fully, then undo around ? (² → √, % → ×100/base); powers → same base, add exponents

TrapAnswer ?, not ?² (?² = 81 → 9); ∛1331 = 11 but √1331 is not whole; ? − 23.33 = 29 → ? = 52.33

• Rearrange to ax² + bx + c = 0; sum of roots = −b/a, product = c/a
• Split middle: 2 numbers, product a × c, sum b → flip their signs, divide by a
• 2x² − 13x + 21: a × c = 42, sum −13 → −6, −7 → roots 6/2, 7/2 = 3, 3.5
• x² = 529 → ±23; y = √529 → +23 only; x³ = 512 → 8 only
• Compare extremes: smallest x vs biggest y, biggest x vs smallest y (all 4 pairs agree?)
• One equal pair → only ≥, ≤ or no relation, never > or <; overlap → no relation

FastSign rule first: all x roots positive and all y roots negative → x > y at once, no factorising

TrapNot dividing by a ((3y + 13) = 0 → y = −13/3, not −13) or not flipping signs; −4.5 is less than −3

• Age difference between two people never changes — use it first
• t years ago / hence → subtract / add t for EVERY person
• Ratio a : b now → ak, bk; after t yrs (ak + t) : (bk + t) = c : d → cross-multiply
• n people: sum rises by n × t after t years; average rises by just t
• 50% more than B → 1.5B; 50% of B → 0.5B; 100/9 % of A → A/9; 200% of Q → 2Q
• Months ÷ 12: 22 yrs 8 months = 22 2/3 yrs (not 22.8)

FastRatio parts differ by 5, real gap 15 yrs → 1 part = 3 yrs; or backsolve options (ages must be whole)

TrapMixed times: shift each person separately; give the age at the asked time (ago / hence / present)

• Profit % is on CP (unless “of SP” is said); discount % is on MP
• SP = CP × 100 + p/100; at loss CP × 100 − l/100; after discount MP × 100 − d/100
• Mark-up m%, discount d% → SP = CP × (1 + m/100)(1 − d/100); 80% up, 30% off → 1.26 CP
• Successive discounts a%, b% → a + b − ab/100 (40% and 25% → 55%, not 65%)
• Buy n, get k free → effective price × n/n + k; the free item still has a cost
• CP : SP = 20 : 27 → profit 7/20 = 35%; GST goes on the discounted price, not MP

FastTake CP = 100; two cases differ by Rs X → their % gap on CP = X (5% − (−15%) = 20% CP = 1500 → CP 7500)

TrapDiscount taken on CP or profit on SP; rotten fruit: profit is on cost of full quantity bought

• Removing mixture keeps the ratio: R litres of a : b take R × a/a + b of item 1
• Adding a pure item changes only that item; the other item stays fixed
• Alligation: cheaper : dearer = (d − m) : (m − c), m = mean
• Profit given → mean CP = SP ÷ (1 + profit%); discount → SP = MP × (1 − discount%)
• Take out x of Q litres and refill, n times → pure left = Q × (1 − x/Q)ⁿ
• Two jars: add milk to milk, water to water — never add ratios

FastMilk = ak, water = bk; remove in the same ratio, add the poured item, equate to new ratio → 1 equation

TrapMean in alligation must be cost price (undo profit/discount); replacement formula only if same amount out and in

• π = 22/7 unless options show π; check radius vs diameter
• Square a: diagonal a√2, area = d²/2; inscribed circle r = a/2, circumscribed r = a/√2
• Rectangle diagonal √(l² + b²); right triangles 3-4-5, 5-12-13, 8-15-17, 7-24-25
• Lengths × k → area × k², volume × k³ (volume 27 : 64 → radius 3 : 4)
• πr² = 154, 616, 1386, 2464 → r = 7, 14, 21, 28; 2πr = 44, 88, 132 → r = 7, 14, 21
• Cylinder CSA : TSA = h : (h + r); TSA − CSA = 2πr²

FastSame volume: r → k × r means h → h/k² (diameter +25% → h × 16/25 → 36% fall)

TrapSemicircle perimeter includes the diameter (πr + 2r); four walls = 2(l + b)h only; sphere volume ratio = (r ratio)³

• Profit ratio = capital × months for each partner; equal time → ratio of capitals
• Joins after x months → invests 12 − x; leaves after x → x; 2-year business → 24
• Capital changes midway → add pieces: 4000 × 6 + 3500 × 6
• Share = Total × own capital-months ÷ sum of all capital-months
• Salary/commission → take out of total first; charity → from that partner's own share
• Sleeping partner still gets share by capital × time; only a working partner's salary is extra

FastCancel HCF early (45000 : 57600 : 40500 → 50 : 64 : 45); months: (C₁ × t₁)/(C₂ × t₂) = share ratio

Trap“Joined after x months” ≠ months invested (12 − x) — check which is asked; salary before split

• Total work = LCM of days; efficiency = work ÷ days; days never add, efficiencies do
• A alone a days, B alone b days → together ab/a + b days
• Efficiency ratio 3 : 4 → time ratio 4 : 3
• x% more efficient → time × 100/100 + x (50% more → time × 2/3)
• M1D1H1E1/W1 = M2D2H2E2/W2; wages ∝ work done (efficiency × days), not days
• Together T; A takes x days more, B y days more than T → T² = x × y

FastJoin/leave: let total days = T, write each person's working days (T, T − 4, 4…), sum of work = total

Trap“Left 4 days before completion” → worked T − 4 days; in alternate-day work the last cycle may be partial

• Downstream D = b + s; upstream U = b − s
• b = D + U/2; s = D − U/2 → D − U = 2s, not s
• Same distance: time up : time down = D : U (inverse of speeds)
• Round trip d each way: time = d/b + s + d/b − s = 2db/b² − s²
• Dead engine → boat drifts at stream speed s only

FastWrite b and s as multiples of one variable (b = 6s → D = 7s, U = 5s); one equation then gives s

Trap“b is 200% more than s” → b = 3s; give b, s, D or U as asked; fewer facts than unknowns → can't be determined

• SI = P × R × T/100 (same every year); T in years: 18 months = 1.5
• CI amount = P × (1 + R/100)ⁿ; CI = Amount − P; rates change yearly → multiply each factor
• CI − SI (2 yrs) = P × (R/100)²; (3 yrs) = P × (R/100)² × (3 + R/100)
• 2-yr CI % of P = 2R + R²/100; 2nd-year CI alone = P × R/100 × (1 + R/100)
• k times a year: rate R/k, periods = years × k (every 8 months: R × 8/12, 3 periods in 2 yrs)

FastUse the table: 10% for 2 yrs CI = 21% of P; CI − SI (2 yrs) is 1% of P at 10%, 4% of P at 20%

TrapAmount vs interest asked; “CI in 2nd year” ≠ 2-year total; non-annual: change both rate and periods

• Distance = train length + length of what it crosses; convert to m and m/s first
• Pole / standing man / car (no length): L ÷ speed; platform P: (L + P) ÷ speed
• Two trains: (L1 + L2) ÷ (s1 + s2) opposite; (L1 + L2) ÷ (s1 − s2) same direction
• Moving man at m: L ÷ (s + m) opposite; L ÷ (s − m) same direction
• Start together towards each other, after meeting take a, b hrs → s1 : s2 = √b : √a

FastPole time t → speed = L/t; platform length = speed × (platform time − pole time)

TrapOvertaking / same direction → subtract speeds; never drop the train's own length; platform = 2L → distance 3L

• A x% more than B → A = B × 100 + x/100; x% less → B × 100 − x/100
• % more/less: A − B/B × 100 — base = the word after than
• A x% more than B → B is x/100 + x × 100 % less than A (25% more → 20% less)
• Successive a%, b% → a + b + ab/100 (minus for a fall); −10%, −15% → −23.5%
• Spend a%, then b% of rest → left = N × 100 − a/100 × 100 − b/100
• Election: margin = (winner % − loser %) of valid votes; valid = cast − invalid

FastAssume 100: chain through one base (C = 100 → B 25% less = 75 → A 20% more = 90); use fractions

Trap“Of the remaining” ≠ of the total; +20% then −20% = −4%, not 0; 10% + 15% successive ≠ 25%

• Two values (roots, ±)? Compare every value of I with every value of II
• Single number vs single number → >, < or = / no relation; never ≥ or ≤
• Roots: one equal pair, all other pairs one way → ≥ or ≤
• II is a fixed number? Plug it into I's conditions; fits exactly → equal
• Units: km/h × 5/18 = m/s; train time = (train + bridge length) ÷ speed
• Find exactly the asked quantity (e.g. (4/3)Y, bridge length), not a step on the way

FastRead option letters first; don't over-calculate — if II = 60 and I is clearly below 60, stop

TrapLetters swap between sets; the two quantities often turn out equal (ages, work, trains) — check equality

• Sum = Average × Count — write totals before and after, then subtract
• n people, 1 joins, avg up d → newcomer = old avg + (n + 1)d (avg falls → − (n + 1)d)
• n people, 1 leaves, avg up d → leaver = old avg − (n − 1)d
• Replace one (count n same) → new − old value = n × change in average
• Wrong values: correct avg = old avg + (correct − wrong) ÷ n
• Two groups: avg = (n1a1 + n2a2)/(n1 + n2); sizes n1 : n2 = (a2 − A) : (A − a1)

FastNewcomer = new avg + old count × rise (7 at 26.5, up 2.5 → 29 + 7 × 2.5 = 46.5 kg)

TrapUse the new count (10 join 48 → 58); correction with the sign reversed is also in the options

• Sufficient = exactly one value; two values, a range or 'greater than' = not sufficient
• Test I alone, then II alone (forget I); combine only if both fail
• Stem data goes with each statement; one statement's data never enters the other's test
• n unknowns need n independent equations; a ratio or % alone gives no value
• Ratio + one actual value → usually sufficient; sum + difference → sufficient
• Quadratic may give 2 values → sufficient only if one is rejected (negative age/length)

FastCount unknowns vs equations; stop once a unique value is possible — don't finish the arithmetic

TrapMarking 'both' when each alone works; right outcome but wrong letter — option order changes every set

• a : b → values ak and bk (one k per ratio); split T in a : b : c → T × a/a + b + c
• Join ratios: make the common term equal: A:B = 9:10, A:C = 3:4 → 9 : 10 : 12
• Chain: A:B = a:b, B:C = c:d → A:B:C = ac : bc : bd
• x% more → (100 + x) : 100: 50% more → 3 : 2; 110% more → 21 : 10
• Same n added to both → difference unchanged; solve ak + n/bk + n = new ratio
• a : b = c : d ↔ a × d = b × c; mean proportional = √(ab); third proportional = b²/a

FastFractions of one thing: (3/8)P = (2/5)Q → P : Q = 2/5 : 3/8 = 16 : 15; k = total ÷ total parts

TrapSame k before and after a change (use k and m); answering parts instead of values (× k)

• Capacity = LCM of times; rate = capacity ÷ time; outlet / leak = minus rate
• Fill in a, empty in b, both open: b > a → fills in ab/b − a; b ≤ a → never fills
• Two fillers a, b → ab/a + b; fills in x, with leak y → leak empties in xy/y − x
• rate(C) = rate(A+B+C) − rate(A+B); negative → C is an emptying pipe
• 20% less time → time × 0.8 → rate × 1.25; efficiency 3 : 4 → time 4 : 3
• Capacity (litres) = rate (L/min) × time to fill alone; keep minutes and hours consistent

FastSwitch: work done = combined rate × hours; remaining work ÷ remaining hours = last pipe's rate

TrapAdding a pipe made the time longer → it is an outlet; don't forget the minus sign for a leak

• Consecutive step: integers 1; odd or even 2; multiples of k → k
• Equally spaced: average = middle term = first + last/2; odd count → sum = count × middle
• Two-digit 10a + b; reversed 10b + a; difference = 9(a − b), sum = 11(a + b)
• 1 + … + n = n(n + 1)/2; first n odd = n²; first n even = n(n + 1); Σn² = n(n + 1)(2n + 1)/6
• HCF × LCM = product of the two numbers; unit digit of 2, 3, 7, 8 repeats every 4; 4, 9 every 2
• Divisible: 3/9 → digit sum; 4 → last 2 digits; 8 → last 3; 11 → alternate-sum gap 0 or 11k

FastSum ÷ count = middle term (5 odd numbers, sum 195 → middle 39 → 35, 37, 39, 41, 43); build around it

TrapStep 1 used for odd/even/multiples; “second lowest” vs “second highest”; answer the asked %/average, not the term

• D = S × T; km/h → m/s × 5/18; m/s → km/h × 18/5 (36 km/h = 10 m/s)
• Towards each other → add speeds; same direction → subtract
• Meet = gap ÷ relative speed; catch up = lead ÷ (faster − slower)
• Average speed = total distance ÷ total time; equal distances → 2ab/a + b
• Fixed distance: speed 5 : 8 → time 8 : 5; x% faster → time falls x/100 + x × 100%
• Circular track L, start together: meet every L/a + b opposite, L/a − b same way

FastDelayed start: move the early starter alone first, then divide the remaining gap by relative speed

TrapAverage speed ≠ a + b/2; 2 h 15 min = 2.25 h, not 2.15 h; with a stop, add the stop time back

• P = favourable ÷ total (0 to 1); P(not A) = 1 − P(A)
• At least one = 1 − P(none)
• Drawn together = without replacement → use nCr; nC2 = n(n − 1)/2, nC3 = n(n − 1)(n − 2)/6
• Different colours together → multiply: exactly 1 red of 2 = (red C1 × non-red C1) ÷ total C2
• Two dice: 36 outcomes; ways for sum S = 6 − |S − 7| (sum 7 → 6 ways, 6 or 8 → 5)
• 52 cards: 4 suits × 13; 26 red, 26 black; 4 each of K, Q, J, A; 12 face cards

FastTotal first (15 balls, 2 drawn → 15C2 = 105), then 1 − none for “at least one”; cancel early, reduce fully

Trap“At least one” ≠ “exactly one”; dice (1, 2) ≠ (2, 1); “first red, then black” fixes order → step-by-step product

• x exceeds y by 5 / x is 5 more than y → x − y = 5; x is 5 less than y → x = y − 5
• a% of x → ax/100; x is what % of y → (x/y) × 100
• ax + b = c → x = c − b/a; a term moved across “=” changes sign
• Minus before a bracket flips every sign inside it
• Approximate → solve exactly, then pick the nearest option

FastOptions far apart → round early (9.75 ≈ 10 → 5x ≈ 452 → x ≈ 90); options close → solve exactly

TrapSubtraction order follows the words (x − 9.75, not 9.75 − x); “four times x and (x − 9.75)” = 4x + x − 9.75

• Floor/box 1 = bottom unless stated; designation ladder (CEO … Clerk) = treat like floors
• “Just/immediately above” = next slot; “above” alone = anywhere higher
• k boxes between → positions differ by k + 1; “A four boxes above C” → A = C + 4
• Week starts Mon or Sun? 30-day months: Apr, Jun, Sep, Nov; Feb 28/29; rest have 31
• Two dates a month → month by month, earlier date first; age = base year − birth year
• Order: slots → fixed clue → linked clues (Case 1/2) → negative clues kill cases → recheck

FastCount clue: with 8 people, “x before M” puts M at slot x + 1; chain “just below” clues into blocks

Trap“Three boxes between” = difference 4, not 3; “above” ≠ “just above”; check other cases before answering

• Facing north: their right = your right; facing south: their right = your left on paper
• “3rd to the right of Y” = 3 seats away (2 sit between); “k between” → seats differ by k + 1
• “k between X and Y” can be on either side → make both cases
• “Immediate/just right” = next seat; “to the right” alone = anywhere on that side
• Row of n: k-th from left = (n − k + 1)-th from right; “as many left as right” → exact middle
• Order: fixed clues → join shared-person clues into blocks → cases → “not” clues pick the case

FastSeat 1 = leftmost: facing north “k right of Y” = Y + k, facing south = Y − k; build longest chain first

TrapOwn left/right for south-facers; “3rd right” read as 3 between; parallel rows face opposite ways

• Only a few / Few / x% A are B = Some A are B + Some A are not B (≠ “Only A are B”)
• Only A are B → All B are A; “Some A are not B” never reverses; All A are B → Some B are A
• Definite: must be true in every diagram → one valid diagram breaking it = does not follow
• Possibility: follows if one valid diagram makes it true — but NOT if it is already definite
• Possibility fails if it breaks a No, Some-not, Only a few or Only statement
• Either-or: both fail, same 2 terms, Some+No or All+Some-not (never Some+Some-not, All+No)

FastNo “No / Some-not / Only a few / Only / x%” statement → no definite negative conclusion can follow

TrapOnly a few A are B kills “All A can be B”; “All A can never be B” = Some A not B, not “No A is B”

• Follows only if its sign is exactly the combined sign (A > B certain → “A ≥ B” false)
• A ≥ B certain → “A > B” alone and “A = B” alone are both false
• “=” links letters: if L = I, a conclusion about L is a conclusion about I
• Either-or only if neither follows, same pair (or “=” linked), all cases covered, no overlap
• Either-or pairs: ≥ → (>, =); ≤ → (<, =); no relation → (≥, <) or (≤, >)
• “A > B” with “A < B” is never either-or (misses A = B)

FastRead only the signs between the two letters: any direction change → no relation; else any strict → strict

TrapAccepting ≥ when the chain gives >; missing either-or via “=” links; either-or when one already follows

• Facing centre: right = anticlockwise, left = clockwise (circle drawn as seen from above)
• Facing outside: right = clockwise, left = anticlockwise
• “3rd to the right” = 3 seats in that person's right direction → 2 sit between
• k between on one side → n − k − 2 on the other side; always try both sides
• Opposite only when n is even: seat + n/2; “neighbours of G face same way” = the two beside G
• Order: 1st person at bottom → no-case clue → branch on “between”/facing → counts kill cases

FastSeats 0 to n − 1 anticlockwise: centre-facing k-th right = seat + k, left = seat − k; outside: swap

TrapOutside-facers' left/right is reversed; “counted from left of D” the wrong way gives n − 2 − correct count

• One word = one code, same in every sentence; code order is random
• Word in two sentences → its code is the code common to both
• Cross off fixed codes; one word + one code left in a sentence → they match
• Two words only in the same single sentence → “Either X or Y” / “Can't be determined”
• New word (in no sentence) → a fresh code used nowhere else
• “If ‘X Y’ is coded as …” is extra info for that one question only

FastStart with words in 3+ sentences, then shared pairs, then leftovers by elimination

TrapMatching codes by position; one code for an either-or word; reusing an old code for a new word

• Right turn: N → E → S → W (clockwise); left turn: N → W → S → E (anticlockwise)
• Start (0, 0): East +x, West −x, North +y, South −y
• Direction of A from / with respect to B → A − B (stand at B, look at A)
• Only x changes → E/W; only y → N/S; dx = 0 is just North/South, not North-east
• Shortest = √(dx² + dy²); total = sum of all legs; keep √41-type answers unrounded
• Triplets: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 7-24-25, 20-21-29

FastWrite (x, y) for every point as you read; then each question is one subtraction

TrapTurning from the wrong facing — write the facing before each turn; “A from B” ≠ “B from A”

• Only relation words fix gender — not “child”, “sibling” or a name; paternal = via father
• Brother-in-law = spouse's brother, or sister's husband
• Sister-in-law = spouse's sister, or brother's wife
• “Only son” still allows daughters; “only child” = no other children
• “No single parent” → every child's father and mother are in the family; find missing spouse
• Use given counts (members, couples, generations, females) to place leftover people

FastOne generation per row, “=” for couples, +/− for gender; for “A to B” walk from B to A, then A's gender

TrapAnswering the reverse relation (S to T ≠ T to S); assuming gender; “Can't be determined” too early

• EJOTY = 5, 10, 15, 20, 25; opposite of letter n = 27 − n (A↔Z, M↔N)
• Pair: gap of positions in word = difference of alphabet values
• “Both directions” → up or down the alphabet counts; “forward only” → must go up in word order
• Repeated letters count separately; each pair counted once
• Meaningful word: find every word; more than one → “more than one” option; none → “none”
• Shift next +1, previous −1; judge vowel on the original letter; keep repeats when sorting

FastWrite alphabet values under letters; compare each only with letters to its right (gap 1 → diff 1 …)

TrapStopping at FILE (LIFE also works); option letter for “more than one”/“none” changes every question

• Row of N: k-th from left = (N − k + 1)-th from right; convert all to from left first
• m-th left of k-th from left = (k − m)-th from left; m-th right of it = (k + m)-th
• m-th right of k-th from right = (k − m)-th from right
• Middle of positions p and q = (p + q) ÷ 2; p + q odd → no single middle
• Preceded by = just left; followed by = just right; £, =, +, @, # … are symbols
• Digit changes: odd/even judged on the original digit; write the new number in full first

FastOne left-to-right scan: stop only at the target type, check its left then right neighbour

TrapPreceded/followed mixed up; counting after dropping without rewriting the row; “within” vs “numbers”

• 8 seats: corner, middle alternate → same type never neighbours; opposite = same type (seat + 4)
• Pattern A: corners face centre, middles out; Pattern B: middles face centre, corners out
• Facing centre: right = anticlockwise; facing outside: right = clockwise
• “Q faces W” = opposite each other, both facing centre; an outside-facer faces nobody
• k-th right/left with k odd → other seat type; k even → same seat type
• Order: “faces”/“opposite” clues → “immediate” clues (other type) → “between” both cases

FastSeats 0–7 anticlockwise, even = corner, odd = middle; centre-facing k-th right = seat + k, outside = seat − k

TrapCentre-facing rule used for an outside-facer; two corners as neighbours; “Q faces W” read as “next to W”

• A1 … Z26; EJOTY = 5, 10, 15, 20, 25; multiples of 3: C F I L O R U X
• Opposite letters add to 27: A–Z, D–W, G–T, J–Q, M–N
• Pair: difference of values = difference of positions (5, 3 two places apart → pair)
• Forward and backward pairs both count; each pair counted once
• Odd/even = digit value, unless “place”/“positioned” → position from the left
• Vowels: A E I O U; next letter = +1 (D → E), previous letter = −1

FastWrite the new number under the old one, digit by digit, then tally; odd one out: write gaps (+4, −2)

TrapEven digit ≠ even position; “twice” vs “more than twice” vs “not repeated”; counting only forward pairs

• Total = position from left + from right − 1 (same for rank from top and bottom)
• From other end = n − p + 1; “k ranks above” = rank number k less
• Opposite ends: a + b ≤ n → between = n − (a + b); a + b > n → between = a + b − n − 2
• “Only k above X” → X is (k + 1)th from top; “more than only k” → (k + 1)th from bottom
• “Not more than one lighter than A” → A lightest/2nd lightest; “not immediately” → 1+ between
• Ladder 1 = heaviest at top; after a swap, A takes B's old position (from both ends)

FastPossible value: strictly between the two neighbours' values; reject options equal to a known value

TrapForgetting − 1 in total; “more than only three” read as “more than three”; not checking a + b > n