Unit 3 of 4
Math That Feels Like Magic
- 22 chapters
- Video 3, chapters 1 to 22
This video explains number tricks, patterns and shapes that seem to work by magic, each by a hidden rule of place value, counting, area, motion or the shape of a surface.
The 1089 trick, Kaprekar's constant, 142857 and multiplication by halving come down to place value, remainders and binary. Pascal's triangle, Fibonacci numbers, the Ulam spiral and sums of odd numbers turn number rules into pictures. A4 paper keeps its shape when halved, and the theorems of Pick, Varignon, Viviani and Hippocrates find exact areas and lengths. The ellipse, the brachistochrone, square wheels and Reuleaux triangles are shapes in motion, and the last chapters run from Archimedes' spherical bands and spherical triangles to Euler's polyhedral formula and the Möbius strip.
Every chapter will be part of one of 4 long videos. The videos are coming soon, and each chapter page already has the definitions and an example for every key term.
Numbers with a hidden mechanism
The 1089 Trick
How the 1089 trick sends every three-digit number with different first and last digits to 1089, explained by place value, digit reversal and the nine three-digit multiples of 99.
Video 3, chapter 1
Kaprekar's Constant and the Road to 6174
How Kaprekar's routine, the largest arrangement of four digits minus the smallest, brings every four-digit start with at least two different digits to Kaprekar's constant 6174, a fixed point, when leading zeros are kept.
Video 3, chapter 2
142857, the Number That Rotates
Why the cyclic number 142857 keeps its six digits in the same circular order when multiplied by 1 to 6, explained by the long division of 1 by 7, its remainder cycle and the period of a repeating decimal.
Video 3, chapter 3
The Repeated Number Trick
Why the repeated number trick works, how digit concatenation multiplies a three-digit block by the number 1001, and how divisibility and the prime factorization of 1001 into 7, 11 and 13 return the block.
Video 3, chapter 4
Multiplication by Halving
How Russian peasant multiplication finds a product by halving and doubling, why the rows with an odd left entry give the answer, and how binary representation, powers of two and ancient Egyptian multiplication explain it.
Video 3, chapter 5
When numbers become pictures
Pascal's Triangle, a Pattern Factory
How Pascal's triangle is built by adding the two entries above, and why its entries are the binomial coefficients that count combinations and lattice paths and give the coefficients in the binomial theorem.
Video 3, chapter 6
Fibonacci Numbers and the Golden Ratio
Why ratios of neighboring terms of the Fibonacci sequence converge to the golden ratio, proved with an update rule for the ratios, and how the square and arc drawing differs from the golden spiral, a logarithmic spiral.
Video 3, chapter 7
The Ulam Spiral, Seeing the Primes
How the Ulam spiral makes prime numbers line up along diagonals, why each diagonal follows a quadratic polynomial, and why no prime-generating polynomial, not even n squared plus n plus 41, gives primes forever.
Video 3, chapter 8
Odd Numbers Build Squares
Why the sum of the first n odd numbers is always n squared, shown by a proof without words that builds square numbers from L-shaped borders of dots, and how the same picture leads to the difference of squares.
Video 3, chapter 9
Geometry that keeps something unchanged
Why A4 Paper Hides a Square Root
How the A-series paper sizes such as A4 get their aspect ratio, the square root of 2, the only ratio for which halving a sheet gives similar rectangles, and why real sheets are rounded to whole millimeters.
Video 3, chapter 10
Pick's Theorem, Area by Counting Dots
How Pick's theorem finds the area of a simple lattice polygon by counting lattice points inside it and on its boundary, why each condition matters, and how a triangulation explains the formula.
Video 3, chapter 11
Varignon's Theorem, a Parallelogram Inside Every Quadrilateral
How the triangle midsegment theorem proves Varignon's theorem, that joining the side midpoints of a quadrilateral gives a parallelogram, and why this Varignon parallelogram has half the area of a convex quadrilateral.
Video 3, chapter 12
Viviani's Theorem, Three Distances and One Total
Why the perpendicular distances from any point inside an equilateral triangle to its three sides always add up to its height, explained by Viviani's theorem and an area decomposition.
Video 3, chapter 13
Hippocrates' Lunes, Crescents That Equal a Triangle
Why two crescents built on the sides of a right triangle have exactly the triangle's area, explained by the lune of Hippocrates, semicircle areas, the Pythagorean theorem and squaring the circle.
Video 3, chapter 14
Shapes in motion
The Ellipse That Aims for You
Why a ball shot from one focus of an elliptical table always bounces through the other focus, explained by the reflection property of the ellipse, the foci of an ellipse, the law of reflection and elliptical billiards.
Video 3, chapter 15
The Brachistochrone, the Fastest Descent
Why a longer curved track can beat a straight one, explained through the brachistochrone curve, the cycloid, conservation of energy and the calculus of variations, for a bead sliding without friction.
Video 3, chapter 16
Square Wheels on the Right Road
How a square wheel can roll with its axle perfectly level on a road of inverted catenary arches, explained through the square wheel problem, the catenary, the hyperbolic cosine and arc length.
Video 3, chapter 17
Reuleaux Triangles and Constant Width
How the Reuleaux triangle, built from three circular arcs, has the same width in every direction, explained with supporting lines, the width of a convex shape and curves of constant width, and why its center still bobs.
Video 3, chapter 18
Curved space and connected surfaces
Archimedes' Spherical Bands
Why bands of equal height on a sphere have equal area, explained by Archimedes' hat-box theorem, the circumscribed cylinder, the area of a spherical zone and the surface area of a sphere.
Video 3, chapter 19
A Triangle with Three Right Angles
How a spherical triangle with great-circle sides can have three right angles, explained through great circles, geodesics, the angle sum of a triangle and the spherical excess in Girard's theorem.
Video 3, chapter 20
Euler's Polyhedral Formula
Why vertices minus edges plus faces equals two for every convex polyhedron, explained by Euler's polyhedral formula, the Euler characteristic, planar graphs and spanning trees.
Video 3, chapter 21
The Möbius Strip, One Side and One Edge
Why the Möbius strip, a band joined after one half-twist, has one side and one edge, explained through non-orientable surfaces, boundary components and topology, and why a center cut leaves one two-sided band.
Video 3, chapter 22