James Tanton

A Dozen Proofs That 0 = 1

Let’s solve in an unusual way. Firstly, is not zero (it is clearly not a solution) so we may divide through by to obtain x =−1−1 x . Substituting into the original equation shows that we can solve instead x 2 −1−1 x +1= 0, that is, x 2 = 1 x , or equivalently And this is easy: is the solution. Inserting this back into the equation shows 1 +1+1= 0, that is, 3 = 0. Dividing by three yields 1 = 0. s t s

A Dozen “Elementary” Questions

Iremember as a fifth-grader asking my teacher, “Why is negative times negative positive?” The response I received was telling. “It just is. Now carry on with the worksheet.” A similarly jarring event occurred to me in grade nine when I questioned the validity of Pythagoras’s theorem. As students, we were asked to construct right triangles on paper, measure their sides with a ruler, and check to see if a2 + b2 equals c2, as they say. I realized then that a finite collection of examples did not constitute proof, although I did not articulate it that way at the time. “Had anyone checked a right triangle the size of the solar system, for instance?” was my thought. I publicly questioned this “check and believe” approach to matters, and my teacher, in response, said to me, “Go back and draw some more right triangles.” I was clearly a troublesome student.

The Isoperimetric Problem

A ccording to lege~d, in the year 800 BC Princess Dido of Tyre fled her Phoenician homeland to free herself of the tyranny of her murderous king brother. She crossed the Mediterranean and sought to purchase land for a new home upon the shores of northern Africa. Confronted with only prejudice and distrust by the natives she was given permission to purchase only as much land as could be surrounded by a hulls hide. She accepted the terms and made the most of them by cutting a hide into very thin strips and piecing them together to form a single, very long string, which she then used to enclose a portion of land of maximal area against the straight line of the coast. The portion of land she consequently purchased for a minimal price turned out to be large enough to build a city. This is said to be the origin of the city of Carthage. Princess Dido, in this legend, solved a variant of the famous isoperimetric problem: Of all figures in the plane with a given perimeter, which encloses the largest area? This is perhaps the oldest problem in global differential geometry. Virgil (70-19 BC) in his work Aeneid refers to the legend of Dido, and the problem, as well as its solution, was certainly known to the Greeks. Throughout the ages many mathematicians have studied the problem offering new approaches and simplified solutions, and for a long while the problem was thought completely solved. In the following twelve problems I lead you through an approach essentially due to Jacob Steiner (1796-1863), with some new twists and tidbits thrown in along the way. Although each puzzle is of interest in its own right, the collection as a whole leads to a thoroughly accessible, and apparently complete, solution to the isoperimetric problem. But there is a warning, one first pointed out by Weierstrass in 1870, that appears in question 12. We will take heed of this warning, but for now, read on, think hard and have fun.

A Dozen Questions about a Triangle

A triangle possesses remarkable properties! For example, the three medians of any triangle always meet in a point, as do the perpendicular bisectors of the sides, and the three altitudes of the triangle. Moreover, as noted by Euler, these three points of intersection are collinear! In 1899, Frank Morley discovered that adjacent pairs of angle trisectors always meet at the vertices of an equilateral triangle (see [1], chapter 12). Any triangle can be dissected into four congruent triangular pieces all similar to the original triangle (the analogous result in three dimensions for tetrahedra, alas, is not true) and any triangle can be used to tile the entire plane. In 1928, E. Spemer used triangular subdivisions to prove that any continuous map from a triangle to itself (interior included) must have a fixed point (Brouwers fixed point theorem). The remarkable properties of a triangle have not escaped the notice of your dozenal correspondent! Here, for your amusement, are twelve problems about our familiar three-sided friend. Enjoy!

A Dozen Questions About the Powers of Two

E veryone is familiar with the powers of two: 1, 2, 4, 8, 16, 32, 64, 128, and so on. They appear with surprising frequency throughout mathematics and computer science. For example, the number of subsets of a finite set is a power of two, as too is the sum of the entries of any row of Pascals triangle. (Mathematically, these two statements are the same!) The largest prime number known today is one less than a power of two, a cube of tofu can be sliced into a maximum of2n pieces with n planar cuts, and every even perfect number is the sum of consecutive integers from 1 up to one less than a power of two! Here I have put together a dozen curiosities all about the powers of two. These puzzles toy with results and ideas from classic number theory and geometry, game theory, and even popular TV culture (one problem is about a variation of the game Survivor)! I hope you enjoy thinking about them as much as I did.