Interlude I

Today is my one hundred and eleventh birthday: I am eleventy-one today! Bilbo Baggins, in J.R.R. Tolkien’s Lord of the Rings With a new-found ability to subtract at speed, you can impress friends with a “magic” trick that, as it’s based on math, never fails (this is but one example of “mathemagic,” about which many books have been written...

Interlude V

I’m sure the universe is full of intelligent life. It’s just been too intelligent to come here. ARTHUR C. CLARKE (reproduced from an interview http://www.scifi.com/transcripts/aclarke.txt) The Vietnam War, during which American casualties ran extremely high, remains controversial in the United States. During the conflict, US forces estimated the strength of enemy forces based on the “SWAG” principle. At the war’s end, in a legal case, Colonel John Stewart took the stand. Lawyers grilled him, asking what, exactly, SWAG stood for. His reply, generating much amusement in the courtroom, was “Scientific Wild-Ass Guess.”...

Arithmetical Advice

This chapter presents advice on how to avoid simple mistakes when performing mental calculations at high speed. It includes a method to speed up the rate at which you recite your multiplication tables. This can save fractions of a second, which, in an exam with many such multiplications, can be crucial. It urges neat handwriting, and shows the superfluity of zeros at the end, or decimal points in the middle of a number, provided you make estimates before calculating an answer. It presents a quick look at factors, which can slash seconds from the time it take to multiply and divide, and introduces the art of shunting.

Calculations with Constraints: Multiply and Divide by Numbers with Specific Properties

This chapter presents the techniques need to multiply or divide two numbers that have a particular property. It also introduces concepts such as twin primes and powerful numbers, and shows the classic proof that there are infinitely many primes. Applications include the size of sheets of paper, the radioactivity of bananas, the dimensions of a rugby union playing field, and the units of time and length proposed by Thomas Jefferson. The methods shown here apply mostly to numbers below, or slightly above 100. As they depend on properties of numbers, there is a need to employ pattern-spotting skills; some hints are provided.

Accounting for Taste: Calculating Columns Quickly

This chapter presents methods by which to add up columns exactly, or to estimate them, more swiftly. As a consequence, these methods can be used in practical situations, such as trips to the local grocery store where you wish to keep an eye on the bill, or a cricket match, where you want to calculate the score. Examples include the number of gifts in “The Twelve Days of Christmas,” the average ages of US presidents, and the number of attendees of the Field of the Cloth of Gold. Also included is the methods used by Gauss to sum the first 100 digits, and Benford’s Law, regarding the distribution of first digits.

Interlude III

Write it on your heart that every day is the best day of the year. No man has learned anything rightly, until he knows that every day is Doomsday. RALPH WALDO EMERSON, Society and Solitude: Twelve Chapters “What day is it?” asked Pooh. “It’s today,” squeaked Piglet. “My favorite day,” said Pooh....

Multiplying and Dividing Irrationally

This chapter describes how to multiply and divide, albeit approximately, by some of the world’s most famous irrational numbers, such as π‎, Euler’s number e, 2, 3, both of which occur frequently in the study of triangles, and the Golden Ratio, also sometimes called the Divine proportion. The approximations for π‎ stem from the Ancient World, including the Hebrew Bible, Greek, and Babylonian approximations. An example for 2 is provided by the medieval Jewish polymath, Maimonides. By use of various approximations, the sine of some angles can be easily computed, which can impress those with a grasp of elementary trigonometry. Some examples of “almost” formulas are presented.

Close-Enough Calculations: Quick and Accurate Approximations

This chapter describes techniques to estimate division and multiplication by certain numbers. The purpose is to help in test taking. By knowing some ways to approximate answers rapidly, some potential answers can be eliminated, enhancing the chance of choosing the correct answer from the smaller number of remaining options. It presents easy ways to determine whether a number is divisible by 11, 17, and 19. It introduces sphenic numbers, cannonball (or square pyramidal numbers) in relation to the Kepler Conjecture, as well as the Kaprekar number. Examples are drawn from the Azerbaijan Grand Prix and two triple dead heats in dog racing.

Super Powers: Calculate Squares, Square Roots, Cube Roots, and More

This chapter describes how to compute square roots, find cube roots and fifth roots of mystery integers, with amazing rapidity. It introduces the concept of the Steinhaus Cyclus. Practical examples include the calculation of body mass index, body surface area, the conversion of acres to square feet, and from square inches to square centimeters. The methods were developed in the ancient world, as well as in early modern Europe. Examples are drawn from cuneiform tablets of the second millennium BCE. The techniques for finding the root of a cubic and fifth power lend themselves to “street” mathematics, or performing mathematics in front of an audience.

Interlude II

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. (God made the integers, all else is the work of man.) LEOPOLD KRONECKER Master, we saw one casting out devils in thy name, and he followeth not us. MARK 9:38 (KING JAMES VERSION)...