394 pie chart

The letter count of each word in this sentence matches

the first 13 digits of the decimal expansion π.

In 1897 amateur mathematician Dr. E. J. Goodwin,

of Solitude, Indiana, thought he had discovered a

remarkable relationship between the circumference of a

circle and its area. Rather than share his discovery with

the mathematical community, Goodwin took his “dis-

covery” to his representative in the Indiana General

Assembly and persuaded him to introduce the follow-

ing house bill:

Be it enacted by the General Assembly of the

state of Indiana, that it has been found that a

circular area is equal to the square on a line

equal to the quadrant of the circumference.

Clearly no politician understood the mathematics of

this assertion, for the bill passed and Goodwin was

well on his way to persuading the state of Indiana that

the area of a circle of circumference Cis given by

, or, equivalently, that the value of πis exactly 4.

Fortunately, due to press attention and the outrage of

mathematicians living in the state, the bill was derailed

before it reached full senate approval and was made into

law. Senator Hubbell, an opponent of the bill who also

saw the absurdity of the claim, aptly remarked: “The

senate might as well try to legislate water to run uphill.”

See also

E

; E

ULER

’

S FORMULA

;

RADIAN MEASURE

;

SCALE

.

pie chart (pie graph) See

STATISTICS

:

DESCRIPTIVE

.

pigeonhole principle (Dirichlet’s principle, drawer

principle, letterbox principle, schubfachprinzip)Attri-

buted to P

ETER

G

USTAV

L

EJEUNE

D

IRICHLET

(1805–59),

the pigeonhole principle observes that if nobjects are

put in pboxes, and n> p, then at least one box receives

two or more objects. The example of housing npigeons

in ppigeonholes is often cited. This principle can also be

stated in the following alternative form:

If each of nobjects is assigned one of plabels,

and n> p, then at least two objects have the

same label.

Thus, for example, if we pull out three socks from a

drawer that contains only black socks and white socks

(three objects, two labels), at least two of those socks

will be the same color.

This straightforward principle has a number of sur-

prising consequences. For instance:

There exist two (nonbald) people in Sydney,

Australia, with exactly the same number of

hairs on their heads.

(Label each person by the number of hairs on his or

her head. It is known that no human has more than 1

million hairs, and there are more than this many peo-

ple in Sydney.)

Any collection of 18 integers contains two inte-

gers that differ by a multiple of 17.

(Assign to each integer its remainder when divided by

17. Noting that there are 17 possible remainders—0

through 16—any collection of 18 integers must contain

two members with the same remainder. Their difference

is a multiple of 17.)

In any group of six people, there are two who

have an identical number of friends in the group.

(Assign to each person the number of friends he or she

possesses in the group. The six numbers 0 through 5

are possible, but not both 0 and 5 simultaneously.

Thus, among the six in the group, only five labels are

possible. Two people must possess the same value.)

The pigeonhole principle can be generalized as

follows:

If each of nk + 1 objects is assigned one of n

labels, then at least k+ 1 objects are assigned

the same label.

(If each label is used at most ktimes, then at most nk

objects are labeled. As there are more objects than this,

some labels must be used more than ktimes.) As conse-

quences we have, for instance:

Among 2,000 people, at least five were born

on the same day of the year.

(Label each person with the day on which he or she

was born. As there are 366 possible labels and 2,000 =

366 ×5 + 170 > 366 ×5 + 1, at least five people have

the same label.)