Arithmetic novelties : When is a number divisible by 3 or 9? ; When is a number divisible by 11? ; Divisibility by prime numbers ; Squaring numbers quickly ; Squares and sums ; Using squares to multiply arbitrary number ; An alternative method for extracting a square root sensible number comparisons ; Euclidean algorithm to find the GCD ; Sums of positive integers ; Sums of odd positive integers ; The realm of nonterminating decimals ; Atoms in the universe of numbers ; Fun with number relationships ; Friendly numbers ; Palindromic numbers ; Prime numbers ; Infinite primes ; The neglected triangular numbers ; Perfect numbers ; Making mistaken generalizations ; The Fibonacci numbers
Algebraic explanations of accepted concepts : Simple algebra helps logical reasoning ; Division by zero ; Irrationality of the square root of 2 ; Bisection method to approximate square roots ; Continued fractions of square roots ; Fermat's method of factoring ; Comparing means ; Diophantine equations ; Falling squares ; Descartes's rule of signs ; Horner's method to evaluate polynomials ; Generating Pythagorean triples ; The Frobenius problem
Geometric curiosities : Parallelograms and triangles ; Using a grid to calculate areas ; The center of a quadrilateral ; Beyond the formula for the area of a triangle ; Heronian triangles ; A new formula for the area of isosceles triangles ; Pick's theorem ; When intersecting lines meet a circle ; Origins of trigonometry ; Sines of small angles ; an unconventional view of the sine ; Surprising proofs of the Pythagorean theorem ; Beyond the Pythagorean theorem: part I ; Beyond the Pythagorean theorem: part II ; Beyond the Pythagorean theorem: part III ; The Pythagorean theorem extended to three dimensions ; Polyhedra: sides, faces, and vertices ; Lunes and the right triangle ; Concurrency ; Similarity and the golden ratio ; A relation between points and circles ; Constructions with compasses alone ; The sphere and the cylinder ; Regular polygons and stars ; Platonic solids and star polyhedra
Probability applied to everyday experiences : How the theory of probability began ; Benford's law ; The birthday phenomenon ; The Monty Hall problem ; Bertrand's box ; The false positive paradox ; Pascal's triangle ; Random walks ; The poker wild-card paradox
Common sense from a mathematical perspective : The origins of some mathematics symbols ; The counterintuitive ; A surprising solution ; Don't "wine" over this problem: a problem-solving approach ; Organized thinking ; Successive percentages ; Rule of 72 ; A mathematical conjecture ; Unexpected patterns ; An infinity conundrum ; The concept of infinity ; Counting the uncountable ; Mathematics on a bicycle ; The parabola: a remarkable curve
Appendix : Ceva's theorem.
