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Verified Quantitative Formulas, Golden Grammar Rules & Static GK Master Points (1990 — 2026 TCS Shift Archive)
112 Quantitative LaTeX Formulas, 100 Golden Rules of English Grammar, and 59 Static GK & Computer Awareness modules verified against 37-Year TCS official question papers.
Quant Formulas
100 Grammar Rules
GK & IT Modules
Speed Math & Option Elimination ArsenalKCR / TOPPER METHODS
Fraction-to-Percentage Wheel (1/1–1/25), Live Digital Root (Cast out 9s) Verifier, & Unit Digit Cyclicity.
The sign method allows instant identification of quadratic root signs without factorizing. For ax² + bx + c = 0: (1) If middle sign is + and constant is +, both roots are negative. (2) If middle is - and constant is +, both roots are positive. (3) If constant c is negative, roots have opposite signs.
Forgetting to divide factors by the coefficient of x² or y² when leading coefficients differ.
The difference between Compound Interest and Simple Interest over 2 years arises entirely from interest on the first year's interest. For 3 years, the additional terms yield D₃ = D₂ × (3 + R/100).
Using the 2-year formula when compounding is half-yearly (for half-yearly compounding over 1 year, use n = 2 half-years and rate R/2).
The rule of alligation is a modified form of weighted average. When a volume x of pure liquid is repeatedly drawn from initial volume V and replaced with water n times, the remaining pure liquid is V(1 - x/V)ⁿ.
Confusing the remaining quantity of pure liquid with the quantity of water added. Water added = Initial Volume - Remaining Pure Liquid.
The slope m of line Ax + By + C = 0 is -A/B. Two non-vertical lines are parallel if and only if their slopes are equal (m₁ = m₂). They are perpendicular if and only if the product of their slopes equals -1 (m₁ × m₂ = -1).
Forgetting the negative sign in m = -A/B for standard form Ax + By + C = 0.
When two angles sum to 90°, tan A = cot B = 1/tan B, hence tan A · tan B = 1. Similarly, sin B = cos A, so sin² A + sin² B = sin² A + cos² A = 1.
Forgetting to handle the middle term (tan 45° or sin² 45°) in symmetric series.
Using the Arithmetic Mean - Geometric Mean (AM ≥ GM) inequality on positive quantities: (a·sin²θ + b·csc²θ) / 2 ≥ √(a·sin²θ · b·csc²θ) = √(ab). Hence the minimum value is 2√(ab). If a < b for sin²θ and csc²θ, the minimum value is a + b.
Applying 2√ab blindly when a < b for sine and cosine forms without checking equality achievable condition.
If from an external point P, a tangent PT and a secant line PAB intersecting the circle at A and B are drawn, then PT² = PA · PB. If two secants PAB and PCD are drawn from P, then PA · PB = PC · PD.
Using AB instead of PB. The formula requires the FULL secant length PB = PA + AB.
Where M is number of workers, D is number of days, H is working hours per day, E is relative worker efficiency, and W is total units of work completed (or wages earned, or consumption consumed).
Putting wages or units produced in the numerator instead of denominator.
Derived directly from sec²θ - tan²θ = 1 = (secθ + tanθ)(secθ - tanθ). Adding both gives 2secθ = x + 1/x => secθ = (x² + 1)/(2x). Subtracting gives 2tanθ = x - 1/x => tanθ = (x² - 1)/(2x). Dividing tanθ by secθ yields sinθ = (x² - 1)/(x² + 1). The same rule applies to cscθ + cotθ = y <=> cscθ - cotθ = 1/y => cosθ = (y² - 1)/(y² + 1).
Forgetting that secθ + tanθ > 1 in the first quadrant, so x is typically greater than 1.
When x + 1/x = k: x² + 1/x² = k² - 2; x³ + 1/x³ = k³ - 3k; x⁴ + 1/x⁴ = (k² - 2)² - 2; x⁵ + 1/x⁵ = (k² - 2)(k³ - 3k) - k; x⁷ + 1/x⁷ = (x³ + 1/x³)(x⁴ + 1/x⁴) - (x + 1/x). When x - 1/x = k: x² + 1/x² = k² + 2; x³ - 1/x³ = k³ + 3k.
Forgetting to subtract (x + 1/x) when multiplying powers to reach x⁵ or x⁷.
Gives the shortest distance d from point P(x₁, y₁) to line Ax + By + C = 0. The coordinate triangle area formula computes the enclosed polygon area from 3 vertices.
Forgetting to normalize coefficients A and B before taking difference |C₁ - C₂| for parallel lines.
For any n-sided regular polygon, the sum of all exterior angles is always 360°. Sum of one interior angle and one exterior angle at any vertex is always 180°. The number of unique diagonals connecting non-adjacent vertices is n(n - 3)/2.
Using the cumbersome interior angle equation when exterior angle gives n directly.
Variance is the mean of squared deviations from the arithmetic mean x̄. Standard deviation σ is the positive square root of variance. Coefficient of variation (CV) measures relative dispersion: a lower CV means greater consistency/stability.
Thinking that adding a constant increases standard deviation. Shift in origin does NOT alter dispersion.
D₂ is the difference between CI and SI for 2 years on Principal P at annual rate R%. For 3 years, D₃ equals D₂ multiplied by (300 + R)/100. Ratio of D₃ / D₂ is always (300 + R) / 100 = 3 + R/100, which allows instant calculation of R without knowing P!
Confusing 3rd year interest difference with 3 years total difference.
Where Q₀ is the initial quantity of pure liquid, V is total capacity/volume, x is quantity removed & replaced with water in each operation, and n is number of operations performed.
Confusing "Milk : Water ratio" with "Milk : Total Mixture ratio". Always use Total in the denominator.
Number of trailing zeros in N! is governed by the exponent of 5, since 2s are abundant. Continue dividing until 5ᵏ exceeds N.
Forgetting power of 25 contributing two 5s, 125 contributing three 5s.
In ΔABC, if AD is a median bisecting side BC at D, the sum of squares of two sides equals twice the sum of squares of the median and half the third side.
Applying angle bisector theorem instead of Apollonius theorem when median is specified.
Where R is bottom base radius, r is top base radius, h is vertical height, and l is slant height of the frustum (bucket shape).
Squaring (R + r) in volume formula: it is (R² + r² + Rr), NOT (R + r)².
For moderately asymmetrical/skewed distributions, Karl Pearson empirical formula links Mode, Median, and Mean.
Swapping coefficients: writing 2 Median - 3 Mean is incorrect and reverses the sign.
Where d is the distance between centres, and r₁ and r₂ are the radii of the two circles. DCT always exists if circles do not lie inside one another. TCT exists only if d ≥ (r₁ + r₂).
Using (r₁ - r₂) instead of (r₁ + r₂) for Transverse Tangents causes instant -0.5 negative penalty.
If (a + b + c) = 0, then a³ + b³ + c³ = 3abc. Also, if a² + b² + c² = ab + bc + ca, then a = b = c.
Forgetting the minus sign: (x - 1/x = k) leads to (k³ + 3k), NOT (k³ - 3k).
For expressions like a·sin²θ + b·cosec²θ, a·cos²θ + b·sec²θ, or a·tan²θ + b·cot²θ, the Minimum value is always 2√(a·b) when a > 0, b > 0.
For a·sin²θ + b·cosec²θ when a < b, the minimum is (a + b), NOT 2√(ab).
If chords AB and CD intersect at P (internally or externally), PA · PB = PC · PD. If PT is a tangent from external point P and PAB is a secant, PT² = PA · PB.
Writing PA · AB instead of PA · PB is the #1 mistake candidates make.
Where I = Incentre (intersection of angle bisectors), O = Circumcentre (perpendicular bisectors), and H = Orthocentre (altitudes).
Confusing Incentre angle (90° + A/2) with Circumcentre angle (2A).
Where s = (a + b + c)/2 is semi-perimeter, r is inradius, R is circumradius, and Δ is the area of the triangle.
Forgetting that r = Δ/s applies to ANY triangle (scalene, isosceles, equilateral).
For a cyclic quadrilateral where all 4 vertices lie on a circle, opposite angles sum to 180°. Product of diagonals equals sum of products of opposite sides.
Applying Brahmagupta's formula to non-cyclic quadrilaterals produces incorrect areas.
An angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides. AD is the internal bisector of ∠A.
Assuming angle bisector is perpendicular to opposite side (only true in isosceles/equilateral).
Where R is base radius, r is top radius, h is vertical height, and l is slant height.
Using (R² + r²) instead of (R² + r² + Rr) in volume loses 2.5 marks.
Total surface area of solid hemisphere includes the circular top (2πr² + πr² = 3πr²). When n smaller spheres of radius r are melted to form a sphere of radius R: R³ = n · r³.
Using 2πr² instead of 3πr² for TOTAL surface area of a solid hemisphere.
This identity is heavily tested in TCS SSC Tier 1 and Tier 2 papers. Given (x² + xy + y²) and (x² - xy + y²), adding them gives 2(x² + y²), subtracting gives 2xy.
Forgetting that (x² + y²) is the arithmetic mean of the two factors.
If A + B = 45° or 225°, then (1 + tan A)(1 + tan B) = 2 and (cot A - 1)(cot B - 1) = 2.
Forgetting the 1/4 factor in the sin and cos triple-angle product formulas.
Standard ratio multipliers eliminate the need to write full tan θ equations. If angles of elevation of top of tower from two points at distances a and b are complementary, Height h = √(ab).
Measuring distances a and b from each other instead of from the base of the tower.
Where φ(n) = n(1 - 1/p₁)(1 - 1/p₂) is Euler's totient function. Wilson's Theorem: (p - 1)! + 1 is divisible by prime p, i.e., (p - 1)! ≡ -1 ≡ (p - 1) (mod p).
Applying Fermat's theorem when the base and divisor are not coprime.
To test divisibility by 72: Last 3 digits must be divisible by 8, and sum of all digits must be divisible by 9. For 88: Last 3 digits div by 8, and (Sum of odd places - Sum of even places) div by 11.
Finding multiple values of y in divisibility by 8 and picking the minimum when question asks for maximum.
Where p, q, r are prime factors. Odd factors = product of (power + 1) of odd primes only. Even factors = Total - Odd factors = a(b+1)(c+1) where p=2.
Counting 1 as a prime factor when prime factorizing.
Trailing zeroes are created by pairs of (2 × 5). In factorials, 2s are abundant, so the number of trailing zeroes is determined entirely by the highest power of 5 in n!.
Adding the original number n into the sum instead of only summing the quotients.
Where D₂ is difference between CI and SI for 2 years, D₃ is difference for 3 years at annual rate R% on Principal P.
Using D₃ = 3 · D₂ (only true if R=0, actual has extra PR³/100³ term).
For standard annual rates: 5% for 2 yrs = 10.25%, 10% for 2 yrs = 21%, 10% for 3 yrs = 33.1%, 20% for 2 yrs = 44%, 20% for 3 yrs = 72.8%.
Applying SI linear multiplication to CI exponential growth.
When a shopkeeper sells at cost price but uses a faulty weight (e.g. 900g instead of 1000g), profit is earned purely on the goods given to the customer.
Dividing the error by 1000g instead of 900g (actual weight given).
The MP/CP ratio formula solves 80% of SSC Tier 1 discount problems instantly without setting variable equations.
Writing [y / x] instead of [y / (x + y)] in "Buy x Get y Free".
In continued proportion a:b = b:c, b is the mean proportional. In a:b = c:d, d is the fourth proportional.
Mixing up Third Proportional (needs 2 terms) with Fourth Proportional (needs 3 terms).
Where x is total capacity of vessel, y is quantity drawn out and replaced with water each time, and n is number of replacement operations.
Thinking alligation gives ratio of Selling Prices instead of Cost Prices.
Where M = Number of workers, D = Days, H = Hours per day, E = Efficiency, W = Work done (or wages earned, food eaten, trenches dug).
Putting wages/work done in the numerator instead of denominator.
When Inlet Pipe A fills in a hours and Outlet/Leak B empties in b hours (where b > a), net filling time = (ab) / (b - a).
Letting the cycle overshoot full tank capacity and then subtracting overfilled water.
Where s₁ and s₂ are two speeds, and Δt is the difference in arrival times (e.g. 10 min late vs 5 min early = 15 min difference).
Forgetting to convert time in minutes to hours (divide by 60) when speeds are in km/h.
Use (+) in relative speed when trains move in OPPOSITE directions. Use (-) when moving in the SAME direction. Speed conversion: 1 km/h = 5/18 m/s, 1 m/s = 18/5 km/h.
Multiplying by 18/5 instead of 5/18 when converting km/h to m/s.
Where D = Downstream Speed (u + v), and U = Upstream Speed (u - v). Total round trip time for distance d: T = d/(u+v) + d/(u-v) = (2du) / (u² - v²).
Adding stream speed in upstream instead of subtracting.
Slope of line ax + by + c = 0 is m = -a/b. Two lines are parallel if m₁ = m₂. Two lines are perpendicular if m₁ · m₂ = -1.
Forgetting the negative sign in the slope formula of general line m = -a/b.
Karl Pearson's empirical formula relates Mode, Median, and Mean for moderately skewed distributions. Standard Deviation σ = √(Variance).
Writing Mode = 2Median - 3Mean instead of 3Median - 2Mean.
Nth term of AP: T_n = a + (n-1)d. Nth term of GP: T_n = a · r^(n-1). Sum of first n natural numbers = n(n+1)/2. Sum of squares = n(n+1)(2n+1)/6. Sum of cubes = [n(n+1)/2]².
Applying S_∞ formula when common ratio r ≥ 1 (series diverges).
Relates the sides of a triangle ABC to the length of median AD drawn to side BC. Also: 3(AB² + BC² + CA²) = 4(m_a² + m_b² + m_c²).
Using Apollonius for angle bisectors instead of medians.
When two chords AB and CD intersect at P (inside or outside the circle), PA · PB = PC · PD. If PT is a tangent from external point P and PAB is a secant, then PT² = PA · PB.
Multiplying PA · AB instead of PA · PB.
For any triangle with area Δ and semi-perimeter s, inradius r = Δ/s and circumradius R = abc/(4Δ). For an equilateral triangle of side a: r = a/(2√3), R = a/√3 (R = 2r).
Using s as perimeter instead of semi-perimeter (a+b+c)/2.
Where R is bottom radius, r is top radius, h is vertical height, and l is slant height. Total Surface Area (TSA) = πl(R + r) + πR² + πr².
Using (R² + r²) in volume instead of (R² + r² + Rr).
For a right prism, cross-sections are identical throughout. For a right pyramid, slant height l = √(h² + r²) where r is inradius of the base.
Using vertical height h instead of slant height l for Lateral Surface Area of a pyramid.
Where p, q, r are distinct prime factors of N. Number of odd factors = (b+1)(c+1) where p=2 is omitted. Number of even factors = a(b+1)(c+1). Product of all factors = N^(Total Factors / 2).
Forgetting to prime-factorize completely before applying exponents (e.g. 12 = 2² · 3, NOT 4 · 3).
Euler's Totient φ(N) counts integers up to N coprime to N. By Euler's Theorem, remainder of a^k divided by N = remainder of a^(k mod φ(N)) divided by N.
Using Euler's theorem when a and N are not coprime (gcd > 1).
To find the unit digit of N^k, divide power k by 4. If remainder is r (1, 2, 3), unit digit = (unit digit of N)^r. If remainder is 0, use power 4.
Using power 0 when power is exactly divisible by 4; power must be 4.
Direct 10-second shortcuts to evaluate 5th and 7th power symmetric polynomials without full expansion.
Adding (x + 1/x) at the end instead of subtracting.
Condition for two equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 to have BOTH roots common: a₁/a₂ = b₁/b₂ = c₁/c₂. Condition for ONE common root: (c₁a₂ - c₂a₁)² = (a₁b₂ - a₂b₁)(b₁c₂ - b₂c₁).
Forgetting the negative sign in sum of roots: α + β = -b/a.
High-speed reduction formula for trigonometric products where angle doubles in each successive term.
Forgetting the 1/4 multiplier in sin/cos triple angle products.
Ratio method eliminates 90% of height and distance algebra in TCS exams. If distance between two elevation angles α and β from opposite sides of tower of height h is d, h = d / (cot α + cot β).
Using d / (√3 + 1) when moving towards tower instead of (√3 - 1).
Where C is total capacity of vessel, x is quantity drawn out and replaced with water in each operation, and n is number of times operation is performed.
Dividing by the remaining quantity instead of total capacity C in (1 - x/C).
If a person travels at S₁ and is t₁ minutes late, and at S₂ is t₂ minutes early: ΔT = (t₁ + t₂)/60 hours. Distance is directly computed in 5 seconds.
Forgetting to convert minutes ΔT to hours by dividing by 60.
Two trains start at the same time from stations A and B towards each other. After crossing each other, they take T₁ and T₂ hours to reach B and A respectively.
Writing S₁/S₂ = √(T₁/T₂) instead of inverse √(T₂/T₁).
On a circular track of length L: Time to first meet = L / (Relative Speed). S₁ - S₂ if running in same direction; S₁ + S₂ if running in opposite directions. Time to meet at starting point = LCM of individual lap times.
Using relative speed for meeting at starting point instead of LCM of lap times.
Where P is principal borrowed, x is equal annual installment paid at end of each year for n years at annual compound rate R%.
Confusing CI installment with Simple Interest installment formula P = nx + [Rx·n(n-1)] / 200.
Fill pipes take positive work; leak/drain pipes take negative work. In alternate hour cycles, calculate work completed before the last positive filling burst to avoid overshoot.
Letting the cycle run past full capacity and applying emptying work when cistern is already full.
Banker's Discount (BD) is simple interest on face value. True Discount (TD) is simple interest on Present Worth (PW). Banker's Gain (BG) is simple interest on True Discount.
Treating True Discount as discount on Face Value instead of Present Worth.
When a dealer sells at cost price but uses a false weight of w grams instead of 1000g, Profit % = [(1000 - w) / w] × 100%.
Dividing error by 1000g instead of the actual false weight delivered to customer.
Wages are divided in proportion to the total work done by each individual (Efficiency × Time), NOT simply by number of days worked unless efficiencies are equal.
Dividing wages in inverse ratio of days when workers worked for different number of days.
In triangle ABC with median AD (length mₐ) drawn to side BC (length a), the sum of squares of the other two sides equals twice the square of the median plus twice the square of half the base.
Forgetting the factor 2 outside the parenthesis: writing b² + c² = mₐ² + (a/2)²
Yields the length of any cevian d from vertex A to base a dividing side BC into segments m and n (where m + n = a).
Swapping m and n with the opposite adjacent sides b and c.
In a cyclic quadrilateral ABCD, the product of the lengths of the two diagonals equals the sum of the products of opposite sides.
Applying Ptolemy's theorem to non-cyclic quadrilaterals (where AC · BD ≤ AB · CD + BC · AD by Ptolemy's inequality).
Where d is the distance between the circumcentre (O) and incentre (I) of a triangle, R is the circumradius, and r is the inradius. This also proves Euler's inequality: R ≥ 2r.
Using R² + 2Rr instead of R² - 2Rr, which would give a distance greater than R.
The area of a triangle formed by the three medians as its sides is equal to 3/4 of the original triangle area. Hence, Area(ΔABC) = (4/3) × Heron's Formula applied to medians.
Multiplying by 3/4 instead of 4/3 to get the area of the original triangle.
The angle formed by the internal bisectors of adjacent angles ∠A and ∠B in quadrilateral ABCD equals half the sum of the remaining angles (∠C + ∠D).
Subtracting angles instead of adding when dealing with adjacent angle bisectors.
Where R is bottom base radius, r is top base radius, h is vertical height, and l is slant height.
Using (R² + r² + 2Rr) instead of (R² + r² + Rr) in volume formula.
Where R is sphere radius, h is cap height, and a is base radius of the cap (a² = h(2R - h)).
Using 4/3 π h³ instead of the exact cap formula.
A regular tetrahedron consists of 4 congruent equilateral triangles of edge a. Ratio of circumradius to inradius R : r = 3 : 1.
Confusing tetrahedron volume (a³ / (6√2)) with square pyramid volume.
Where s = (a + b + c + d)/2 is semi-perimeter of cyclic quadrilateral ABCD. If quadrilateral is also tangential (a+c = b+d), Area = √(abcd).
Applying Brahmagupta's formula to non-cyclic quadrilaterals without the cos²(θ/2) Bretschneider correction term.
For cubic polynomial ax³ + bx² + cx + d = 0 with roots α, β, γ: Sum of roots S₁ = -b/a; Sum of pairwise products S₂ = c/a; Product of roots S₃ = -d/a.
Forgetting alternating signs: -b/a, +c/a, -d/a, +e/a.
Allows instant computation of αⁿ + βⁿ without solving for the roots individually. For cubic ax³ + bx² + cx + d = 0: a Sₙ + b Sₙ₋₁ + c Sₙ₋₂ + d Sₙ₋₃ = 0.
Setting S₀ = 0 instead of S₀ = α⁰ + β⁰ = 1 + 1 = 2.
Standard recurrence for odd powers of (x + 1/x). For x⁵ + 1/x⁵ = (k² - 2)(k³ - 3k) - k.
Forgetting to subtract (x + 1/x) after multiplying lower powers.
A parabola opens upward if a > 0 (minimum at vertex) and downward if a < 0 (maximum at vertex). Vertex coordinates are (-b/(2a), -D/(4a)).
Assuming a quadratic with negative a has a minimum value (it tends to -∞).
Key identity: sin θ · sin(60°-θ) · sin(60°+θ) = (1/4) sin 3θ; cos θ · cos(60°-θ) · cos(60°+θ) = (1/4) cos 3θ; tan θ · tan(60°-θ) · tan(60°+θ) = tan 3θ.
Including 1/4 factor for tan product (tan product has NO 1/4 factor!).
Calculates the sum of sines or cosines when angles form an AP with common difference β and n terms.
Using nβ instead of nβ/2 in numerator angle.
When lines/wires are drawn from the tops of two vertical poles of heights h₁ and h₂ to the feet of each other, their intersection point is at height h above the ground, regardless of the distance between poles!
Thinking height h depends on the distance d between the two poles.
Euler's Totient φ(n) counts the positive integers up to n coprime to n. If gcd(a,n)=1, the remainder when a^k is divided by n is a^(k mod φ(n)) mod n.
Applying Euler's theorem when gcd(a, n) ≠ 1.
A natural number p > 1 is a prime if and only if (p - 1)! + 1 is divisible by p. Corollary: (p - 2)! divided by prime p always leaves remainder 1.
Applying Wilson's theorem to composite divisors.
Where d(N) = (a+1)(b+1)... is the number of factors, σ(N) is the sum of factors, and Product of factors = N^(d(N)/2).
Including 2⁰ (which is 1) when computing the sum of even factors.
In successive division, the quotient obtained in the first step is divided by the second divisor, and that subsequent quotient is divided by the third divisor.
Dividing the original number N repeatedly instead of dividing successive quotients.
When different quantities x₁, x₂, ..., xₙ are successively removed from total capacity C and replaced by another liquid (or water), the pure content multiplier is ∏(1 - xᵢ/C).
Using the reducing remaining volume in denominator instead of constant total capacity C.
On a circular track of length L: Time to meet at start is LCM of individual lap times. Time to first meet is L / (relative speed). Number of distinct meeting points on track = (a - b) in same direction, (a + b) in opposite direction where a:b is coprime speed ratio.
Confusing meeting at starting point (LCM of lap times) with meeting anywhere on track.
In a race of distance D: "A gives B a start of x metres" means B starts at distance x ahead (B runs D - x). "A beats B by t seconds" means when A finishes at time T, B takes time (T + t).
Simply adding headstarts x + y instead of subtracting the fractional overlap (xy/D).
Where N is total stationary visible steps. If walking in direction of moving escalator: Total Steps = Visible Steps Taken + Escalator Steps. In opposite direction: Total = Steps Taken - Escalator Steps.
Adding escalator speed when walking against the motion of the escalator.
When equal distances are covered at speeds v₁, v₂, ..., vₙ, average speed is the Harmonic Mean of speeds, NOT the arithmetic mean.
Taking simple arithmetic mean (v₁ + v₂ + v₃)/3 when distances are equal (arithmetic mean only applies when TIMES are equal).
In a correct clock, the two hands coincide every 65 5/11 minutes (720/11 min). If hands coincide every M minutes: If M < 720/11, clock is GAINING; if M > 720/11, clock is LOSING.
Using 65 minutes instead of 65 5/11 (720/11) minutes for normal coincidence interval.
When arranging n distinct items in a circle where clockwise and anticlockwise orderings are distinct (e.g. people at round table), ways = (n-1)!. When arrangement can be flipped over (necklaces, beads, garlands), ways = (n-1)! / 2.
Dividing by 2 for table seating arrangements where looking from above distinguishes clockwise from counterclockwise.
A derangement is a permutation where none of the n elements appears in its original/natural position. Recurrence: Dₙ = (n - 1)(Dₙ₋₁ + Dₙ₋₂).
Confusing derangements with normal permutations (n!).
Calculates the posterior probability of event Aᵢ given that evidence B has occurred. Base probabilities P(Aᵢ) are updated using conditional likelihoods P(B|Aᵢ).
Confusing P(A|B) (probability of A given B) with P(B|A) (probability of B given A).
For two circles with radii r₁ and r₂ separated by center distance d: The Direct Common Tangent (DCT) length is √(d² - (r₁ - r₂)²). The Transverse Common Tangent (TCT) length is √(d² - (r₁ + r₂)²). Since (r₁ + r₂)² > (r₁ - r₂)², DCT is always strictly longer than TCT.
Accidentally swapping the signs: subtracting radii for TCT instead of adding radii.
From an external point P, if PT is a tangent touching the circle at T, and PAB is a secant intersecting the circle at A and B, then PT² = PA · PB. If two secants PAB and PCD are drawn from P, then PA · PB = PC · PD. If two chords AB and CD intersect inside a circle at E, then AE · EB = CE · ED.
Multiplying PA by AB instead of the whole secant segment PB = PA + AB.
In △ABC with incenter I, excenter E opposite to A, circumcenter Cₒ, and orthocenter H: Incenter angle ∠BIC = 90° + A/2; Excenter angle ∠BEC = 90° - A/2; Circumcenter angle ∠BCₒC = 2A; Orthocenter angle ∠BHC = 180° - A. Notice: ∠BIC + ∠BEC = 180° (supplementary).
Mixing up the orthocenter (180° - A) and circumcenter (2A) formulas.
In a trapezium with parallel sides a, b and oblique sides c, d: Area = 1/2(a + b)h. The sum of squares of diagonals is d₁² + d₂² = c² + d² + 2ab. For a rhombus with side a and diagonals d₁, d₂: Area = 1/2 d₁ d₂, and 4a² = d₁² + d₂² because diagonals bisect each other at right angles.
Using a² = d₁² + d₂² for rhombus instead of 4a² = d₁² + d₂².
Master Calendar Formula: (A + B + C + D + E) mod 7, where A = Century Code (4, 2, 0, 6 for remainder 1, 2, 3, 0 when century digits divided by 4); B = Last 2 digits of year; C = Quotient of B / 4 (leap years); D = Month Code (Jan=0, Feb=3, Mar=3, Apr=6, May=1, Jun=4, Jul=6, Aug=2, Sep=5, Oct=0, Nov=3, Dec=5); E = Date. Day codes: 0=Sun, 1=Mon, 2=Tue, 3=Wed, 4=Thu, 5=Fri, 6=Sat.
Forgetting to use Jan=6 and Feb=2 when dealing with dates in January or February of a LEAP year.
Angle between hands at H hours and M minutes is given by θ = |30H - (11/2)M|. Relative speed between minute and hour hand is 5.5° per minute (11/2°/min). Mirror image time in a 12-hour clock = 11:60 - Given Time. Water image time = 18:30 (or 17:90) - Given Time.
Subtracting from 12:00 instead of 11:60 for mirror time, which leads to arithmetic borrow mistakes.
1. Standard Die: Sum of opposite faces = 7 (1<->6, 2<->5, 3<->4). Adjacent faces NEVER sum to 7. 2. General Die Clockwise Rule: Identify the single common face and write digits clockwise to identify opposites. 3. Painted Cube Cutting: For a cube cut into n parts along each edge: 3-faces painted = 8 (corners), 2-faces = 12(n-2) (edges), 1-face = 6(n-2)² (faces), 0-faces = (n-2)³ (core).
Assuming a die is standard when two adjacent visible faces add up to 7 (e.g. showing 3 and 4 adjacent). That is a general die!
Key Figure Counting Invariants: 1. Vertically divided triangle with n sectors: Total = n(n+1)/2. 2. Square with both diagonals and internal rays intersecting at center: Total = 2n, where n is the count of smallest individual triangles. 3. 5-pointed Star = 10 triangles (5 small outer + 5 large base triangles). 4. Number of squares in n x n grid = 1² + 2² + ... + n².
Counting squares as non-rectangles (every square is mathematically a rectangle).
Core Ranking Invariants: 1. Total persons = Left Rank + Right Rank - 1. 2. Rank from Right = Total - Rank from Left + 1. 3. Position Interchange: Total = (New Rank of person A) + (Old Rank of person B) - 1. 4. Overlapping condition (Minimum students): Total_min = (L + R) - (Middle + 2). Non-overlapping (Maximum students): Total_max = L + R + Middle.
Forgetting the "- 2" in the overlapping formula T_min = (L + R) - (between + 2).
Displacement calculation uses the Pythagoras theorem: d = √(Δx² + Δy²). Shadow Rules: At Sunrise, Sun is in East => Shadows fall to the WEST. If facing South, shadow is to your RIGHT. If facing North, shadow is to your LEFT. At Sunset, Sun is in West => Shadows fall to the EAST. At 12:00 Noon, Sun is overhead => No shadow.
Mixing up shadow direction at sunrise vs sunset.
Alphabetical indexing invariants: 1. Forward Position + Reverse Position = 27 (e.g. A=1, Z=26 => 1+26=27). 2. EJOTY Anchor: E(5), J(10), O(15), T(20), Y(25). 3. CFILORUX Anchor: Multiples of 3 (C=3, F=6, I=9, L=12, O=15, R=18, U=21, X=24). 4. Opposite Pairs: A-Z, B-Y, C-X, D-W, E-V, F-U, G-T, H-S, I-R, J-Q, K-P, L-O, M-N.
Subtracting from 26 instead of 27 to find reverse alphabetical rank.