Colossal Gardner, ch. 2

Chapter 2 is a continuation of the section on Arithmetic and Algebra.

This time, we get the Calculus of Finite Differences. This concept was first published between 1715 and 1717 by Brook Taylor, who developed the Taylor Theorem. Finite differences are an early form of integral calculus, and the idea is to look at the results of an equation in steps, effectively taking the differences of that equation at those steps to determine the underlying properties of an event. It’s not an infallible way of building up equations to explain specific behavior, but it’s a good starting point in many cases.

Gardner describes a “mind reading” party trick used by W. W. Sawyer, who taught math at Wesleyan University. You ask someone to think of a quadratic equation, with no powers greater than x^2 (to keep things simple).

For example, use 5*x^2 + 3*x – 7.

You ask the person to give you the answers for x = 0, 1 and 2. You can turn your back on them so you can’t see them working the math. In this case, the answers would be -7, 1 and 19. With practice, you could do the following steps in your head in a couple seconds.

The process is to write out the answers in a row:
-7 1 19

Subtract the number on the left from its neighbor to the right and put the results on the next row:

(1 – -7) = 8, and (19 – 1) = 18
8 18

And do the same thing to build up the third row.

18 – 8 = 10

-7 1 19
8 18
10

The original equation can be built up by the following rules:
The coefficient for x^2 is 1/2 of the bottom number.
The coefficient for x is the first number of the middle row minus half the bottom number.
The constant is the first number of the top row.

a*x^2 + b*x + c
a = 10/2 = 5
b = 8 – 10/2 = 3
c = -7

The calculus of finite differences is a study of these kinds of equations. It had been further developed by Leonhard Euler and George Boole (creator of boolean logic) but fell out of favor in the 1800’s. In the later 1900’s, it became useful again for use in statistics and the social sciences. You can determine the gravitational constant this way. Time the fall of a stone and record the distances at one second intervals:

0 16 64 144 256
16 48 80 112
32 32 32

distance = 32/2 * x^2 + (16 – 32/2) + 0 = 16*s^2

There’s no guarantee that this equation holds absolutely in all cases, but it’s one way to start approaching the development of a theory.

I’d done something like this myself many years ago when I was bored one day, looking at the differences of different strings of powers:

1 4 9 16 25 36
3 5 7  9  11
2 2  2  2

1 8 27 64 125 216
7 19 37 61 91
12 18 24 30
6  6  6

So, I’d discovered finite differences without having any idea what it was. Note that as you get higher powers of x, you get more lines on the pyramid.

Gardner gives a couple additional puzzles to play with – What’s the maximum number of pieces a pancake can be cut into by n straight cuts? How many different designs can you get when making circular necklaces of n beads of two colors (the beads are white or black). And, what’s the maximum number of triangles that can by made with n straight lines?

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