Recreational mathematics

Math That Surprises You

Famous paradoxes, puzzles, number magic and fractals, each a named surprise followed by the mathematics that explains it.

  • Introductory
  • 91 chapters
  • Free
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About this course

Math That Surprises You consists of four standalone long videos about paradoxes, puzzles, mathematics that looks like magic, and fractals. Every chapter starts with a named surprise, such as the Monty Hall problem, the mutilated chessboard, Kaprekar’s constant or the Koch snowflake, and then works out the mathematics that explains it. Each video introduces the ideas it uses, so you can start with any video or any chapter, and the outline marks the few chapters that build on an earlier one.

Paradoxes That Fool You goes from misleading pictures and everyday quantities to chance, choices that go in circles and infinity, and ends with what the Banach-Tarski theorem actually says. Puzzles That Make You Think solves classic puzzles with invariants, planned moves, binary codes, shared reasoning, game strategies and graphs. Math That Feels Like Magic explains number tricks, number patterns drawn as pictures, exact areas and lengths in geometry, curves in motion and surfaces such as the Möbius strip. Fractals, Patterns Without End builds fractals from repeated rules, measures them with fractal dimension and follows iteration into chaos, Julia sets, the Mandelbrot set and Newton fractals.

The videos are coming out one at a time, starting with Paradoxes That Fool You. Every chapter has its own page with precise definitions, a worked example for each key term and a few questions with answers, so you can start reading now. School arithmetic with fractions and percentages is enough to begin. Many chapters also use school algebra and geometry, a few are easier with basic probability, and logarithms, binary numbers and complex numbers are explained where they are used. The chapter on Newton fractals also uses the derivative of a polynomial.

Other courses on the site go further into the mathematics, and none of them is needed here. Introduction to Probability covers conditional probability, Bayes’ theorem, the birthday problem and expected value, and Statistics treats Simpson’s paradox with two-way tables and weighted averages. Calculus develops geometric series and repeating decimals, volumes by cross-sections, the cycloid, the catenary and Newton’s method. Linear Algebra I covers linear transformations and complex numbers, the language behind the maps of iterated function systems, and Differential Equations treats bifurcations, chaos and the Lorenz system.

Good to know first

  • School arithmetic, fractions and percentages
  • School algebra and geometry help in many chapters
  • Basic probability helps for a few chapters

Course outline

23 of 91 chapters are on YouTube so far. Each chapter is part of one of 4 long videos. Open a unit to see its chapters, or follow a chapter for its definitions and examples.

  1. Paradoxes That Fool You

    All 23 chapters on YouTube
    Open the Paradoxes That Fool You unit page

    What the picture hides

    1. The Missing SquareOn YouTube

      Video 1, chapter 1

      missing square puzzle, dissection puzzle, slope and collinear points

    2. The Coin Rotation ParadoxOn YouTube

      Video 1, chapter 2

      coin rotation paradox, rolling without slipping, circumference, line of centers and frame of reference

    3. Aristotle's WheelOn YouTube

      Video 1, chapter 3

      Aristotle's wheel paradox, concentric circles, rolling and sliding and angular speed

    4. The Napkin Ring ParadoxOn YouTube

      Video 1, chapter 4

      napkin ring problem, Cavalieri's principle, cross-section, annulus and volume of a sphere

    5. The Potato ParadoxOn YouTube

      Video 1, chapter 5

      potato paradox, percentage, percentage point and ratio

    6. The Missing DollarOn YouTube

      Video 1, chapter 6

      missing dollar riddle, double counting and net payment

    What random really means

    1. Monty HallOn YouTube

      Video 1, chapter 7

      Monty Hall problem, conditional probability, sample space and switching strategy

    2. Bertrand's Box ParadoxOn YouTube

      Video 1, chapter 8

      Bertrand's box paradox, Bayes' theorem, equally likely outcomes and selection procedure

    3. The Birthday ParadoxOn YouTube

      Video 1, chapter 9

      birthday problem, complement rule, number of pairs and multiplication rule

    4. Simpson's ParadoxOn YouTube

      Video 1, chapter 10

      Simpson's paradox, weighted average, aggregated data and confounding variable

    5. The Friendship ParadoxOn YouTube

      Video 1, chapter 11

      friendship paradox, size-biased sampling, average degree and social network

    6. The Waiting-Time ParadoxOn YouTube

      Video 1, chapter 12

      waiting-time paradox, inspection paradox, length-biased sampling and expected waiting time

    7. Bertrand's Chord ParadoxOn YouTube

      Video 1, chapter 13

      Bertrand's chord paradox, random chord, geometric probability and probability model

    When better goes in circles

    1. Nontransitive DiceOn YouTube

      Video 1, chapter 14

      nontransitive dice, transitive relation, win probability and expected value

    2. Penney's GameOn YouTube

      Video 1, chapter 15

      Penney's game, coin toss sequence, overlapping patterns and state diagram

    3. Braess's ParadoxOn YouTube

      Video 1, chapter 16

      Braess's paradox, Nash equilibrium, traffic network, selfish routing and social optimum

    When infinity changes the rules

    1. Zeno's Achilles and the TortoiseOn YouTube

      Video 1, chapter 17

      Zeno's paradoxes, geometric series, convergent series and relative speed

    2. Why 0.999... Equals 1On YouTube

      Video 1, chapter 18

      repeating decimal, limit of a sequence, real number and finite truncation

    3. Hilbert's HotelOn YouTube

      Video 1, chapter 19

      Hilbert's Hotel, countably infinite set, one-to-one correspondence, proper subset and pairing function

    4. Can Every Fraction Be Counted?On YouTube

      Video 1, chapter 20

      rational numbers, countable set, dense set, lowest terms and diagonal enumeration

      Builds on Hilbert's Hotel.

    5. Cantor's Diagonal ArgumentOn YouTube

      Video 1, chapter 21

      Cantor's diagonal argument, uncountable set, proof by contradiction and decimal expansion

      Builds on Why 0.999... Equals 1 and Can Every Fraction Be Counted?.

    6. The Ross-Littlewood UrnOn YouTube

      Video 1, chapter 22

      Ross-Littlewood paradox, supertask, set-theoretic limit and divergent sequence

    7. Banach-Tarski and What the Theorem Actually SaysOn YouTube

      Video 1, chapter 23

      Banach-Tarski paradox, nonmeasurable set, axiom of choice, rigid motion and cardinality

      Builds on Hilbert's Hotel.

  2. Puzzles That Make You Think

    24 chapters coming soon
    Open the Puzzles That Make You Think unit page

    See what cannot change

    1. The Mutilated ChessboardComing soon

      Video 2, chapter 1

      mutilated chessboard problem, domino tiling, invariant and coloring argument

    2. The Missing-Square Tromino TilingComing soon

      Video 2, chapter 2

      L-tromino, tromino tiling, mathematical induction, divide and conquer and base case

    3. The Impossible Fifteen PuzzleComing soon

      Video 2, chapter 3

      15 puzzle, sliding puzzle, inversion, permutation parity and unsolvable position

    4. Frogs and ToadsComing soon

      Video 2, chapter 4

      frogs and toads puzzle, lower bound, counting argument and minimum number of moves

    Plan the moves

    1. The River-Crossing ShuttleComing soon

      Video 2, chapter 5

      river crossing puzzle, state space, legal state and shuttle strategy

    2. The Bridge and TorchComing soon

      Video 2, chapter 6

      bridge and torch problem, optimal schedule, greedy strategy and exhaustive search

    3. The Water-Jug PuzzleComing soon

      Video 2, chapter 7

      water pouring puzzle, greatest common divisor, multiples and exact measurement

    4. The Tower of HanoiComing soon

      Video 2, chapter 8

      Tower of Hanoi, recursion, recurrence relation and exponential growth

    5. Two Eggs and One Hundred FloorsComing soon

      Video 2, chapter 9

      egg drop problem, worst-case analysis, binary search and triangular numbers

    Extract information

    1. The Counterfeit CoinComing soon

      Video 2, chapter 10

      counterfeit coin problem, balance scale, decision tree and ternary search

    2. The Broken Stick PuzzleComing soon

      Video 2, chapter 11

      broken stick problem, triangle inequality, probability as area and uniform distribution

    3. One Thousand Samples, Ten TestsComing soon

      Video 2, chapter 12

      group testing, pooled samples, binary encoding and bit

    4. The Hat-Parity PuzzleComing soon

      Video 2, chapter 13

      prisoners and hats puzzle, parity and parity bit

    5. The One-Coin Chessboard MessageComing soon

      Video 2, chapter 14

      chessboard coin puzzle, XOR, binary labels and parity check

    Reason together

    1. The Prisoners and the Light SwitchComing soon

      Video 2, chapter 15

      light switch puzzle, counting protocol, distributed algorithm and fairness assumption

    2. The 100 Prisoners PuzzleComing soon

      Video 2, chapter 16

      100 prisoners problem, random permutation, permutation cycle and cycle-following strategy

    3. The Blue-Eyed IslandersComing soon

      Video 2, chapter 17

      blue-eyed islanders puzzle, common knowledge, public announcement and epistemic logic

    4. Knights and KnavesComing soon

      Video 2, chapter 18

      knights and knaves puzzle, truth-tellers and liars, logical consistency and case analysis

    Find the winning move

    1. The Take-Away GameComing soon

      Video 2, chapter 19

      subtraction game, winning position, losing position, backward induction and normal play

    2. NimComing soon

      Video 2, chapter 20

      Nim, nim-sum, Bouton's theorem, binary numbers and combinatorial game

    3. ChompComing soon

      Video 2, chapter 21

      Chomp, strategy-stealing argument, nonconstructive proof and first-player win

    Turn a route into a graph

    1. The Seven Bridges of KönigsbergComing soon

      Video 2, chapter 22

      Seven Bridges of Königsberg, graph theory, Eulerian path, Eulerian circuit and vertex degree

    2. The Icosian Tour and Hamiltonian CyclesComing soon

      Video 2, chapter 23

      Icosian game, Hamiltonian cycle, dodecahedral graph and graph traversal

      Builds on The Seven Bridges of Königsberg.

    3. The Party Handshake PuzzleComing soon

      Video 2, chapter 24

      pigeonhole principle, simple graph, degree sequence and vertices of equal degree

  3. Math That Feels Like Magic

    22 chapters coming soon
    Open the Math That Feels Like Magic unit page

    Numbers with a hidden mechanism

    1. The 1089 TrickComing soon

      Video 3, chapter 1

      1089 trick, place value, digit reversal and multiples of 99

    2. Kaprekar's Constant and the Road to 6174Coming soon

      Video 3, chapter 2

      Kaprekar's constant, Kaprekar's routine, fixed point and leading zero

    3. 142857, the Number That RotatesComing soon

      Video 3, chapter 3

      cyclic number, long division, remainder cycle and period of a repeating decimal

    4. The Repeated Number TrickComing soon

      Video 3, chapter 4

      divisibility, prime factorization, the number 1001 and digit concatenation

    5. Multiplication by HalvingComing soon

      Video 3, chapter 5

      Russian peasant multiplication, ancient Egyptian multiplication, binary representation, halving and doubling and powers of two

    When numbers become pictures

    1. Pascal's Triangle, a Pattern FactoryComing soon

      Video 3, chapter 6

      Pascal's triangle, binomial coefficients, binomial theorem, combinations and lattice paths

    2. Fibonacci Numbers and the Golden RatioComing soon

      Video 3, chapter 7

      Fibonacci sequence, golden ratio, ratio convergence, golden spiral and logarithmic spiral

    3. The Ulam Spiral, Seeing the PrimesComing soon

      Video 3, chapter 8

      Ulam spiral, prime numbers, quadratic polynomial and prime-generating polynomial

    4. Odd Numbers Build SquaresComing soon

      Video 3, chapter 9

      sum of odd numbers, square numbers, proof without words and difference of squares

    Geometry that keeps something unchanged

    1. Why A4 Paper Hides a Square RootComing soon

      Video 3, chapter 10

      A4 paper, A-series paper sizes, aspect ratio, square root of 2 and similar rectangles

    2. Pick's Theorem, Area by Counting DotsComing soon

      Video 3, chapter 11

      Pick's theorem, lattice polygon, lattice points and triangulation

    3. Varignon's Theorem, a Parallelogram Inside Every QuadrilateralComing soon

      Video 3, chapter 12

      Varignon's theorem, Varignon parallelogram, triangle midsegment theorem and quadrilateral

    4. Viviani's Theorem, Three Distances and One TotalComing soon

      Video 3, chapter 13

      Viviani's theorem, equilateral triangle, perpendicular distance and area decomposition

    5. Hippocrates' Lunes, Crescents That Equal a TriangleComing soon

      Video 3, chapter 14

      lune of Hippocrates, Pythagorean theorem, semicircle area and squaring the circle

    Shapes in motion

    1. The Ellipse That Aims for YouComing soon

      Video 3, chapter 15

      reflection property of the ellipse, foci of an ellipse, elliptical billiards and law of reflection

    2. The Brachistochrone, the Fastest DescentComing soon

      Video 3, chapter 16

      brachistochrone curve, cycloid, calculus of variations and conservation of energy

    3. Square Wheels on the Right RoadComing soon

      Video 3, chapter 17

      square wheel, catenary, arc length and hyperbolic cosine

    4. Reuleaux Triangles and Constant WidthComing soon

      Video 3, chapter 18

      Reuleaux triangle, curve of constant width, supporting line and width of a convex shape

    Curved space and connected surfaces

    1. Archimedes' Spherical BandsComing soon

      Video 3, chapter 19

      Archimedes' hat-box theorem, spherical zone, surface area of a sphere and circumscribed cylinder

    2. A Triangle with Three Right AnglesComing soon

      Video 3, chapter 20

      spherical triangle, great circle, geodesic, spherical excess and angle sum of a triangle

    3. Euler's Polyhedral FormulaComing soon

      Video 3, chapter 21

      Euler's polyhedral formula, Euler characteristic, convex polyhedron, planar graph and spanning tree

    4. The Möbius Strip, One Side and One EdgeComing soon

      Video 3, chapter 22

      Möbius strip, non-orientable surface, boundary component, topology and half-twist

  4. Fractals, Patterns Without End

    22 chapters coming soon
    Open the Fractals, Patterns Without End unit page

    One rule, repeated

    1. A Shape Made from Smaller Versions of ItselfComing soon

      Video 4, chapter 1

      fractal, self-similarity, iteration and finite approximation

    2. The Koch Snowflake, a Boundary That Keeps GrowingComing soon

      Video 4, chapter 2

      Koch snowflake, Koch curve, infinite perimeter and finite area

    3. The Cantor Set and What Remains After RemovalComing soon

      Video 4, chapter 3

      Cantor set, ternary expansion, measure zero and open interval

    4. Sierpinski's Triangle, Holes Inside HolesComing soon

      Video 4, chapter 4

      Sierpinski triangle, Sierpinski gasket, recursive removal and zero area

    Different recipes, the same structure

    1. Pascal's Triangle Turns into a FractalComing soon

      Video 4, chapter 5

      Pascal's triangle modulo 2, parity of binomial coefficients and modular arithmetic

      Builds on Sierpinski's Triangle, Holes Inside Holes.

    2. The Chaos GameComing soon

      Video 4, chapter 6

      chaos game, random iteration and attractor

      Builds on Sierpinski's Triangle, Holes Inside Holes.

    3. Iterated Function Systems, a Tiny Recipe for a ShapeComing soon

      Video 4, chapter 7

      iterated function system, contraction mapping, invariant set and uniqueness of the attractor

      Builds on The Chaos Game.

    4. Barnsley's FernComing soon

      Video 4, chapter 8

      Barnsley fern, affine transformation, self-affine set and map probabilities

      Builds on The Chaos Game and Iterated Function Systems, a Tiny Recipe for a Shape.

    5. L-Systems, a Grammar That Grows BranchesComing soon

      Video 4, chapter 9

      L-system, Lindenmayer system, string rewriting, turtle graphics and formal grammar

    Measuring the unfamiliar

    1. The Dragon Curve, Folding a PathComing soon

      Video 4, chapter 10

      dragon curve, Heighway dragon, paperfolding sequence and turn sequence

    2. Fractal Dimension, Between a Line and an AreaComing soon

      Video 4, chapter 11

      fractal dimension, similarity dimension, logarithm, Hausdorff dimension and topological dimension

      Builds on The Koch Snowflake, a Boundary That Keeps Growing, The Cantor Set and What Remains After Removal and Sierpinski's Triangle, Holes Inside Holes.

    3. Box Counting, Measuring a Pattern Across ScalesComing soon

      Video 4, chapter 12

      box-counting dimension, log-log plot, grid resolution and dimension estimate

      Builds on Fractal Dimension, Between a Line and an Area.

    4. Hilbert's Space-Filling CurveComing soon

      Video 4, chapter 13

      Hilbert curve, space-filling curve, continuous surjection and nested squares

    5. The Menger SpongeComing soon

      Video 4, chapter 14

      Menger sponge, three-dimensional fractal, zero volume and connected framework

      Builds on Fractal Dimension, Between a Line and an Area.

    From shapes to changing systems

    1. Coastlines and the Scale of a MeasurementComing soon

      Video 4, chapter 15

      coastline paradox, measurement scale, statistical self-similarity and ruler length

    2. The Logistic Map, Repeating One EquationComing soon

      Video 4, chapter 16

      logistic map, cobweb diagram, stable fixed point, two-cycle and discrete dynamical system

    3. Period Doubling and the Route to ChaosComing soon

      Video 4, chapter 17

      period-doubling bifurcation, bifurcation diagram, deterministic chaos and periodic window

      Builds on The Logistic Map, Repeating One Equation.

    4. The Lorenz Attractor and Order Inside ChaosComing soon

      Video 4, chapter 18

      Lorenz attractor, Lorenz system, butterfly effect, phase space and sensitive dependence on initial conditions

    A map of mathematical worlds

    1. Complex Numbers and the Plane Behind the PicturesComing soon

      Video 4, chapter 19

      complex number, complex plane, imaginary unit, complex multiplication and modulus and argument

    2. Julia Sets, One Rule and Many Starting PointsComing soon

      Video 4, chapter 20

      Julia set, filled Julia set, orbit, escape radius and quadratic iteration

      Builds on Complex Numbers and the Plane Behind the Pictures.

    3. The Mandelbrot Set, a Map of Julia WorldsComing soon

      Video 4, chapter 21

      Mandelbrot set, parameter plane, escape-time algorithm and connectedness

      Builds on Complex Numbers and the Plane Behind the Pictures and Julia Sets, One Rule and Many Starting Points.

    4. Newton Fractals and Where an Approximation Will LandComing soon

      Video 4, chapter 22

      Newton fractal, Newton's method, basin of attraction and cube roots of unity

      Builds on Complex Numbers and the Plane Behind the Pictures, Julia Sets, One Rule and Many Starting Points and The Mandelbrot Set, a Map of Julia Worlds. Also uses derivatives of polynomials.