Logical Reasoning
Opposite Face Sum in Dice — In a standard die, opposite faces always sum to 7 (1-6, 2-5, 3-4). Use this to eliminate wrong answers in cube unfolding problems instantly.
Clock Hands Direction — Hour hand moves 0.5° per minute, minute hand 6° per minute. Angle between them = |11M/2 - 30H| where M=minutes, H=hours. Quick mental check for direction sense.
Mirror Image Letters — Vertically symmetric letters (A,H,I,M,O,T,U,V,W,X,Y) look same in mirror; others flip horizontally. Remember "A HIM OUT VW XY" for quick identification.
Dice Net Adjacent Rule — In a dice net, faces sharing an edge are adjacent, never opposite. Skip one face between to find the opposite face when counting around.
Blood Relations Gender Markers — Mother's/Father's side always determines maternal/paternal. Son/Daughter of Sister = Nephew/Niece; Son/Daughter of Brother also = Nephew/Niece. Uncle's/Aunt's children = Cousins.
Paper Cutting Folds — Each fold doubles the cuts. One fold + one cut = 2 holes; two folds + one cut = 4 holes; n folds + k cuts = k×2ⁿ holes (unless on fold line).
Mathematical Reasoning
Divisibility by 11 — Subtract alternating digits (1st-2nd+3rd-4th...). If result is 0 or divisible by 11, original number is divisible by 11. Example: 121 → 1-2+1=0 ✓
Sum of AP Formula — Sₙ = n/2(2a + (n-1)d) or n/2(first + last). Remember "Half times First-plus-Last" for quick area/sum calculations.
Quadratic Sum-Product Roots — For ax²+bx+c=0, sum of roots = -b/a, product = c/a. Remember "Minus-B-over-A, C-over-A" to write equations from roots quickly.
Circle Tangent Right Angle — Tangent always perpendicular to radius at contact point (90°). Use this with Pythagoras for length calculations in tangent-radius problems instantly.
Heron's Formula Shortcut — A = √[s(s-a)(s-b)(s-c)] where s=(a+b+c)/2. For quick checks, if s=10 and sides 6,8,6 then s-a=4, s-b=2, s-c=4, compute √(10×4×2×4).
Polynomial Remainder Theorem — p(a) gives remainder when p(x) divided by (x-a). To find remainder when dividing by (x-2), just calculate p(2). Zero remainder means (x-a) is factor.
Euclid's Division Lemma — a = bq + r where 0≤r<b. HCF(a,b) = HCF(b,r). Keep dividing until remainder=0; last divisor is HCF. Works for all integer pairs.
Everyday Mathematics
Percentage-Fraction Equivalents — 50%=1/2, 33.33%=1/3, 25%=1/4, 20%=1/5, 12.5%=1/8, 10%=1/10. Memorize these for instant conversions in speed calculations.
Profit/Loss Percentage — Always on Cost Price (CP). SP=CP(100±P%)/100. If two items sold at same SP with x% profit and x% loss, net result is always loss of x²/100 percent.
Simple Interest Formula — SI = PRT/100 (Principal×Rate×Time/100). Amount = P(1+RT/100). For compound: A=P(1+R/100)ⁿ where n=time periods.
Speed-Distance-Time Triangle — Cover 'S', see D/T (Speed=Distance/Time). Cover 'D', see S×T. Cover 'T', see D/S. Draw triangle with D on top, S and T below.
Relative Speed — Same direction: subtract speeds. Opposite direction: add speeds. For circular tracks, relative speed determines meeting time; use LCM for simultaneous returns.
Work Formula Efficiency — If A does work in 'a' days, 1-day work = 1/a. Combined work: 1/a + 1/b = 1/total. Total days = (ab)/(a+b) for two workers together.
Achievers Section
Area of Triangle Coordinate — Area = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|. Cross-multiply diagonally in matrix form, subtract products for quick computation without drawing.
Probability Sum Rule — P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events, last term becomes zero. Always check if events can occur simultaneously.
Mean-Median Quick Check — In symmetric data, Mean=Median=Mode. If Mean>Median, distribution is right-skewed (tail to right). If Mean<Median, left-skewed. Quick data pattern recognition.