210 Math Trivia Questions & Answers

Sharpen your skills with this big, balanced set of math trivia.

You’ll start with friendly warm-ups—primes, shapes, simple equations—and build toward deeper ideas like Fourier series, Gödel’s theorems, and cryptography.

Perfect for classrooms, quiz nights, and curious minds who love numbers, patterns, and problem-solving.

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Numbers & Number Theory

Q: What makes a number prime?
A: It has exactly two positive divisors: 1 and itself.

Q: What is the smallest prime?
A: 2.

Q: Is 1 prime or composite?
A: Neither; it’s a unit.

Q: What is the only even prime?
A: 2.

Q: What’s a composite number?
A: An integer with more than two positive divisors.

Q: What’s the next Fibonacci number after 13?
A: 21.

Q: What is the golden ratio φ equal to (closed form)?
A: (1 + √5) / 2 ≈ 1.618.

Q: What’s the smallest perfect number?
A: 6 (1 + 2 + 3 = 6).

Q: What is the 10th triangular number?
A: 55.

Q: What’s the divisibility test for 3?
A: A number is divisible by 3 if its digit sum is divisible by 3.

Q: gcd(24, 36) equals what?
A: 12.

Q: lcm(6, 15) equals what?
A: 30.

Q: What does Goldbach’s conjecture assert?
A: Every even integer > 2 is the sum of two primes (unproven).

Q: What are twin primes?
A: Primes differing by 2 (e.g., 11 and 13); infinitude is unknown.

Q: Evaluate 17 mod 5.
A: 2.

Q: What is Euler’s totient φ(9)?
A: 6.

Q: Sum of the first n positive integers?
A: n(n + 1)/2.

Q: Who first showed √2 is irrational?
A: Ancient Greeks (often credited to the Pythagorean school).

Q: Name two famous transcendental numbers.
A: e and π.

Q: Mod 3, what residues can perfect squares have?
A: 0 or 1.

Q: What is a Mersenne prime?
A: A prime of the form 2^p − 1 with p prime.

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Algebra & Equations

Q: Solve 2x + 3 = 7.
A: x = 2.

Q: State the quadratic formula.
A: x = [−b ± √(b² − 4ac)] / (2a).

Q: What does the discriminant tell you?
A: b² − 4ac indicates the nature/number of roots.

Q: Factor x² − 9.
A: (x − 3)(x + 3).

Q: Expand (a + b)².
A: a² + 2ab + b².

Q: Solve x + y = 10 and x − y = 4.
A: x = 7, y = 3.

Q: When is a function one-to-one?
A: Distinct inputs give distinct outputs.

Q: Inverse of f(x) = 2x + 5?
A: f⁻¹(x) = (x − 5)/2.

Q: State the product rule for logs.
A: log(ab) = log a + log b (same base).

Q: Solve 3^x = 27.
A: x = 3.

Q: Determinant of \[a,b],[c,d]\[a,b],[c,d]?
A: ad − bc.

Q: Eigenvalues of \[2,0],[0,3]\[2,0],[0,3]?
A: 2 and 3.

Q: Compute (1 + i)².
A: 2i.

Q: Formula for “n choose k.”
A: n! / (k!(n − k)!).

Q: Vieta’s relations for x² − sx + p = 0?
A: Roots sum to s and multiply to p.

Q: Degree of a product of polynomials?
A: Sum of degrees (assuming nonzero polynomials).

Q: Solve |x − 3| = 5.
A: x = 8 or x = −2.

Q: 5th term of a geometric sequence with a₁ = 2, r = 3?
A: 2·3⁴ = 162.

Q: Exponential growth vs. decay?
A: Base > 1 grows; 0 < base < 1 decays.

Q: Solve 5x − 7 < 8.
A: x < 3.

Q: What does the Cayley–Hamilton theorem say?
A: A matrix satisfies its own characteristic polynomial.

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Geometry & Shapes

Q: Sum of angles in a Euclidean triangle?
A: 180°.

Q: State the Pythagorean theorem.
A: a² + b² = c² in a right triangle.

Q: Area of a circle?
A: πr².

Q: Circumference of a circle?
A: 2πr.

Q: Volume of a sphere?
A: (4/3)πr³.

Q: Area of a triangle (base b, height h)?
A: ½bh.

Q: What makes triangles similar?
A: Equal corresponding angles and proportional sides.

Q: Angles of an equilateral triangle?
A: 60°, 60°, 60°.

Q: What’s a regular polygon?
A: All sides and angles equal.

Q: Sum of interior angles of an n-gon?
A: (n − 2)·180°.

Q: Exterior angle of a regular n-gon?
A: 360°/n.

Q: What is a 3-4-5 triangle?
A: A Pythagorean triple; any multiple is also right.

Q: State Heron’s formula.
A: Area = √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2.

Q: Distance between (x₁, y₁) and (x₂, y₂)?
A: √[(x₂ − x₁)² + (y₂ − y₁)²].

Q: Slopes of perpendicular lines (in ℝ²)?
A: Negative reciprocals (when defined).

Q: Name common triangle congruence criteria.
A: SSS, SAS, ASA, AAS, and RHS/HL.

Q: Euler’s formula for convex polyhedra?
A: V − E + F = 2.

Q: How many Platonic solids exist?
A: 5 (tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Q: Where do a triangle’s perpendicular bisectors meet?
A: At the circumcenter.

Q: How does the centroid split each median?
A: In a 2:1 ratio (vertex to centroid : centroid to midpoint).

Q: On a sphere, a triangle’s angle sum is…
A: Greater than 180°.

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Calculus & Analysis

Q: d/dx (x²) = ?
A: 2x.

Q: ∫ xⁿ dx (n ≠ −1) equals?
A: xⁿ⁺¹/(n + 1) + C.

Q: Derivatives of sin x and cos x?
A: (sin x)′ = cos x; (cos x)′ = −sin x.

Q: What is the Fundamental Theorem of Calculus (gist)?
A: Differentiation and integration are inverse processes.

Q: limₓ→0 (sin x)/x = ?
A: 1.

Q: State the chain rule.
A: (f ∘ g)′(x) = f′(g(x))·g′(x).

Q: State the product rule.
A: (uv)′ = u′v + uv′.

Q: Power series for eˣ?
A: Σₙ xⁿ/n! from n = 0 to ∞.

Q: When does Σ rⁿ converge (|r| < 1)?
A: For |r| < 1; sum = 1/(1 − r).

Q: What is a Riemann sum?
A: A finite sum approximating an integral via rectangles.

Q: What’s a critical point?
A: Where f′ = 0 or f′ is undefined.

Q: Mean Value Theorem statement?
A: There exists c with f′(c) = [f(b) − f(a)]/(b − a).

Q: When can you use L’Hôpital’s rule?
A: For 0/0 or ∞/∞ indeterminate forms (with conditions).

Q: What is a partial derivative?
A: A derivative with respect to one variable holding others fixed.

Q: What does the gradient indicate?
A: Direction of steepest increase; magnitude = maximal rate.

Q: What is a double integral?
A: An integral over a 2D region; accumulates “volume” under a surface.

Q: Divergence vs. curl (vector fields)?
A: Divergence measures sources/sinks; curl measures rotation.

Q: What is a Fourier series for?
A: Representing periodic functions as sums of sines/cosines.

Q: Pointwise vs. uniform convergence (idea)?
A: Uniform uses one N for all x; pointwise N can depend on x.

Q: Why Lebesgue over Riemann sometimes?
A: Handles more limits and messy sets via measure theory.

Q: Can a continuous function be nowhere differentiable?
A: Yes (e.g., the Weierstrass function).

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Probability & Statistics

Q: Valid range for probability values?
A: 0 to 1 inclusive.

Q: P(rolling a 6) on a fair die?
A: ⅙.

Q: Expected value of a fair die roll?
A: 3.5.

Q: Mean vs. median?
A: Mean is average; median is middle value.

Q: What is the mode?
A: Most frequent value.

Q: What does standard deviation measure?
A: Spread around the mean.

Q: Law of large numbers (idea)?
A: Sample mean → expected value as n grows.

Q: Bayes’ theorem formula?
A: P(A|B) = P(B|A)P(A)/P(B).

Q: Independence condition?
A: P(A ∩ B) = P(A)P(B).

Q: Mutually exclusive events satisfy…
A: P(A ∩ B) = 0.

Q: What is the binomial distribution?
A: Number of successes in n independent Bernoulli trials.

Q: What’s the 68-95-99.7 rule?
A: For normal data: ≈68%/95%/99.7% within 1/2/3σ.

Q: Define a p-value.
A: Probability of data (or more extreme) given a true null.

Q: What does a 95% confidence interval mean?
A: The procedure captures the parameter in 95% of long-run samples.

Q: What is Simpson’s paradox?
A: A trend reverses when groups are combined.

Q: Optimal Monty Hall strategy?
A: Switch; win rate ≈ ⅔.

Q: Birthday paradox: people for ~50% match?
A: 23.

Q: Expected flips to first heads (p = ½)?
A: 2.

Q: Var(X + Y) for independent X, Y?
A: Var(X) + Var(Y).

Q: Central Limit Theorem (gist)?
A: Standardized sums tend to normal as n grows.

Q: Does correlation imply causation?
A: No; r ranges from −1 to 1.

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Logic, Set Theory & Foundations

Q: Example of set-builder notation for positives?
A: {x ∈ ℝ : x > 0}.

Q: Cardinality of the empty set ∅?
A: 0.

Q: What is the power set P(S)?
A: All subsets of S; |P(S)| = 2^{|S|}.

Q: State De Morgan’s laws (sets).
A: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ; (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.

Q: When is p → q false?
A: Only when p is true and q is false.

Q: Contrapositive of “If p then q”?
A: If ¬q then ¬p (logically equivalent).

Q: Are the natural numbers countable?
A: Yes.

Q: Are the real numbers countable?
A: No; they’re uncountable.

Q: Axiom of Choice is equivalent to what famous results?
A: Zorn’s Lemma and the Well-Ordering Theorem.

Q: What is Russell’s paradox?
A: The set of all sets not containing themselves leads to contradiction.

Q: Gödel’s incompleteness (first theorem)?
A: Any consistent system rich enough for arithmetic is incomplete.

Q: What’s a tautology?
A: A statement true under all valuations.

Q: Propositional vs. predicate logic difference?
A: Predicate logic includes quantifiers/variables.

Q: Symbols for “for all” and “there exists”?
A: ∀ and ∃.

Q: The “barber paradox” illustrates what?
A: Self-reference causing contradiction (a variant of Russell’s).

Q: Do mathematicians include 0 in ℕ?
A: Often yes; conventions vary by context.

Q: Are the rationals countable?
A: Yes (they can be listed).

Q: Continuum hypothesis status in ZFC?
A: Independent (can’t be proved or disproved).

Q: Boolean algebra corresponds to which set operations?
A: AND/OR/NOT ↔ ∩/∪/complement.

Q: What are truth tables for?
A: Systematically evaluating compound statements.

Q: Church–Turing thesis (idea)?
A: “Computable” equals “Turing-computable” (a widely accepted thesis).

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Famous Mathematicians & History

Q: Who wrote Elements?
A: Euclid.

Q: Who approximated π and studied levers?
A: Archimedes.

Q: Who’s often credited with a² + b² = c²?
A: Pythagoras (attribution is traditional).

Q: Who co-invented calculus?
A: Isaac Newton and Gottfried Wilhelm Leibniz.

Q: Who is the “Prince of Mathematicians”?
A: Carl Friedrich Gauss.

Q: Who popularized modern notation (e, i, f(x), Σ)?
A: Leonhard Euler.

Q: Who founded group theory and died at 20?
A: Évariste Galois.

Q: Which algebraist’s theorem links symmetries and conservation laws?
A: Emmy Noether.

Q: Who developed set theory and compared sizes of infinity?
A: Georg Cantor.

Q: Whose incompleteness theorems reshaped logic?
A: Kurt Gödel.

Q: Indian genius famed for partitions and mock theta functions?
A: Srinivasa Ramanujan.

Q: Who posed 23 problems in 1900?
A: David Hilbert.

Q: Father of computer science and codebreaker?
A: Alan Turing.

Q: Who formalized probability axioms and has a complexity concept named after him?
A: Andrey Kolmogorov.

Q: Champion of fractal geometry; namesake set?
A: Benoit Mandelbrot; the Mandelbrot set.

Q: Who proved Fermat’s Last Theorem (1994–1995)?
A: Andrew Wiles.

Q: Who resolved the Poincaré conjecture?
A: Grigori Perelman (2002–2003 preprints; recognized in 2006).

Q: First woman to win the Fields Medal?
A: Maryam Mirzakhani (2014).

Q: Australian polymath and 2006 Fields Medalist?
A: Terence Tao.

Q: Ancient Alexandrian mathematician and philosopher?
A: Hypatia.

Q: Early woman number theorist using the pseudonym “M. LeBlanc”?
A: Sophie Germain.

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Mathematical Puzzles & Recreational Math

Q: 1729 is famous because…
A: It’s the smallest number expressible as two cubes in two ways.

Q: The Bridges of Königsberg problem launched which field?
A: Graph theory.

Q: Minimum moves for n-disk Towers of Hanoi?
A: 2ⁿ − 1.

Q: Minimum moves for 4 disks?
A: 15.

Q: What’s a knight’s tour?
A: A sequence of knight moves visiting every square once.

Q: Magic constant of the 3×3 Lo Shu square?
A: 15.

Q: What’s a Latin square?
A: n×n grid where each symbol appears once per row/column.

Q: Four Color Theorem says…
A: Any planar map needs at most four colors (adjacent regions differ).

Q: What is a pentomino?
A: A shape made of five equal squares edge-to-edge.

Q: Conway’s Life is what kind of system?
A: A cellular automaton with gliders/oscillators.

Q: What’s a repunit?
A: A number like 11, 111—only digit 1 in some base.

Q: Collatz conjecture rule?
A: n → n/2 if even, 3n + 1 if odd (repeat; unproven).

Q: Fifteen puzzle parity fact?
A: Only half of scrambled states are solvable.

Q: Define a magic square.
A: All rows, columns, and main diagonals sum to the same value.

Q: Hamiltonian path vs. cycle?
A: Path visits each vertex once; cycle returns to start.

Q: Eulerian circuit definition?
A: A closed walk using each edge exactly once.

Q: Banach–Tarski paradox claims…
A: A ball can be decomposed and reassembled into two identical balls (using AC).

Q: What algorithm quickly finds gcd(a, b)?
A: The Euclidean algorithm.

Q: What’s a palindromic number?
A: Same digits forward and backward.

Q: What’s a pandigital number?
A: Uses each digit exactly once (often 1–9 or 0–9).

Q: What’s a perfect magic cube?
A: 3D magic square with equal sums in rows, columns, pillars, and main space diagonals.

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Applied Math in the Real World

Q: Compound interest formula (compounded n times per year)?
A: A = P(1 + r/n)^{nt}.

Q: Rule of 72 estimates what?
A: Doubling time ≈ 72 / (interest rate in %).

Q: GPS positioning relies on what?
A: Trilateration from satellite distances.

Q: JPEG compression uses which transform?
A: Discrete Cosine Transform (DCT).

Q: PageRank is fundamentally based on what?
A: Eigenvectors of a link matrix.

Q: RSA’s security rests on what hard problem?
A: Factoring large integers.

Q: Core optimizer in many ML models?
A: Gradient descent (and variants).

Q: PID in control stands for…
A: Proportional–Integral–Derivative.

Q: Epidemiology metric R₀ indicates…
A: Average secondary cases from one infection (R₀ > 1 suggests growth).

Q: Hamming codes do what?
A: Detect/correct bit errors in transmissions.

Q: Linear programming is often solved by…
A: The simplex method (or interior-point methods).

Q: What does queuing theory study?
A: Waiting lines (e.g., M/M/1 models).

Q: A Nash equilibrium is…
A: A profile where no player benefits by unilateral deviation.

Q: Arrow’s impossibility theorem says…
A: No voting rule meets all “fairness” criteria simultaneously.

Q: The traveling salesman problem (TSP) is…
A: NP-hard; exact solutions scale poorly.

Q: Fractal dimension models what phenomenon?
A: Roughness (e.g., coastlines with dimension between 1 and 2).

Q: Nyquist–Shannon sampling theorem states…
A: Sample at ≥ twice the highest frequency.

Q: Runge–Kutta methods are used for…
A: Numerically solving differential equations.

Q: Cryptographic hash functions should be…
A: Preimage- and collision-resistant.

Q: Which stochastic model underlies Black–Scholes?
A: Geometric Brownian motion.

Q: A Kalman filter does what?
A: Fuses predictions with noisy measurements optimally (under assumptions).

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Math Across Cultures & Notation

Q: Write 2025 in Roman numerals.
A: MMXXV.

Q: Which ancient base led to 60 minutes per hour?
A: Babylonian base-60.

Q: The Maya primarily used which base?
A: Base-20 (vigesimal) with a zero symbol.

Q: Our numeral system’s origin and route?
A: Hindu–Arabic numerals from India via Arabic scholars.

Q: Name two abaci.
A: Chinese suanpan and Japanese soroban.

Q: “Algebra” comes from what word?
A: Arabic “al-jabr” (from al-Khwarizmi’s title).

Q: Who introduced the symbol π in print?
A: William Jones (1706); Euler popularized it.

Q: Who set rules for zero and negatives in the 7th century?
A: Brahmagupta.

Q: Who popularized decimal fractions in Europe?
A: Simon Stevin (1585).

Q: Who created logarithms (1614)?
A: John Napier.

Q: A slide rule works because of what property?
A: log(ab) = log a + log b (scales add).

Q: Which revolution birthed the metric system?
A: The French Revolution (1790s).

Q: What were Inca quipu used for?
A: Recording numbers/records with knotted cords.

Q: Who popularized Σ for summation?
A: Euler (widespread use).

Q: Who introduced the equals sign “=”?
A: Robert Recorde (1557).

Q: Who championed binary numeration in the West?
A: Gottfried Wilhelm Leibniz.

Q: What are Penrose tiles notable for?
A: Aperiodic tilings with local rules (1970s).

Q: “Vedic mathematics” is…
A: A 20th-century mental-math system; ancient origin claims are debated.

Q: Which reals have periodic continued fractions?
A: Quadratic irrationals (Lagrange’s theorem).

Q: What were Japanese sangaku?
A: Shrine tablets with challenging geometry problems.

Q: Who introduced the symbol ∈ for set membership?
A: Giuseppe Peano (late 19th century).