## never get outta the boat

Posted by vlorbik on November 16, 2009

i first encountered the factorial function

at about age ten. in fact, i recently

acquired a copy of the very book i

leaned about factorials from.

i blogged about it here,

mentioning the (classic!) problem—”four fours”—

i learned about ’em *from*.

the game is, using only “standard” operations

like powering and rooting and multiplying

and subtracting and whatnot…

and *exactly* four 4’s…

and no other numerals..

to write representations of

small natural numbers.

e.g.

1 = 44/44

2 = 4*4/(4+4)

3 = (4+4+4)/4

4 = 4*4^(4-4)

and so on. a great game for kids.

(you can see it had something of

an influence on *me*…)

kids of all ages lest that go without saying.

anyhow, sooner or later you’ll get stuck.

two things happen. you give up or you

get mad and start looking more carefully.

okay, three. you can *cheat* and allow

“new” symbols… like factorial (!).

the factorial function “counts permutations”.

in the example that should be given

every time the subject come up

until the student indicates that

they’re already doing it “in their

head” every time it come up already

and you can stop again (already):

the permutations of the elements

of {A, L, T} are

ALT, ATL, LAT, LTA, TAL, and TLA.

*any* three letters can be used of course;

the permutations of {X, Y, Z}

(i’m being sloppy) are

XYZ, XZY, YXZ, YZX, ZXY, and YXZ.

the point… *a* point anyway…

is that a set of *three* letters

will always have *six* permutations.

one easily sees that this is “because”

6 = 3*2*1. likewise for {A,B,C,D}

one has 4*3*2*1 permutations.

notation:

4! = 4*3*2*1 = 24

3! = 3*2*1 = 6

2! = 2*1 = 2

1! = 1 = 1

the “factorial of” a (natural) number…

*n*, say…

is denoted by “postfixing”

(like some… trouble aplenty…

adjective-postpositive)

the symbol “!”

(i pronounce this “bang”

usually in class…

“exclamation point”

has five times the

necessary number

of syllables…).

we now introduce the weird-looking

but not-so-weird-if-you-just-look-closer

convention that

0! = 1

(there’s *one* way, from anyway

*one* point of view to “list”

the “elements of” the empty set

[i.e., the set of *no* elements…

the “zero” case of “how many elements?”]:

namely the empty *list*).

we can now (though i consider it highly

optional) define the factorial function

!:N—>N

by

!(0) = 1

!(n) = n*!(n-1) [ n\=0],

a “recursive” definition.

these amuse prepared minds

and horrify the rest.

best not try it on the class as a whole

unless they’ve got some “math maturity”.

really n-factorial is spelled “n!”.

i used !(n)

to be perfectly explicit

about the fact that we

*are* considering a

*function* on N

(the set of natural numbers

[including zero; rant still

to come unless it’s around

here somewhere.

the point is to know like your own middle name

that when you need to count orderings you’ll

*use* this thing (and to know when you *see* it

what the heck it is).

students that can’t write out all 120

permutations of {E,G,B,D,F} at this point,

and go on to the rest of the course anyway,

are *damaged* thereby

and indeed constitute damage to

their whole class and to society at large.

i don’t like this any better than anybody else.

but what i *really* don’t like is being the only

god-damn doctor of philosophy i know of

saying so on the record at this level of detail.

your philosophy is sick and i’m here to fix it.

oh, cursed spite.

you don’t have to be all tough-guy

this-is-college-kid about it… never mind

the if-you-were-serious-you’d-already-*know*

game that wrecks most math classes

before they even get started…

actually, starry-eyed idealist that i am,

i believe that material much easier

than tying your fucking shoes

can probably be taught even

to the dimmest kid admitted

to your college if that’s what

you actually want to fucking do.

i could be wrong of course.

## Sue said

I’ve usually seen this game described as – get all the numbers from 1 to 100 by using 4 fours. Have you done it? I’ve gotten lots of them but not all. A colleague showed me another “cheat”. Hard to write it in the comments section. .4 repeating (ie use a bar on top) can be used to good effect. He’s gotten all 100.

## kibrolv said

i’ll have looked at the solutions provided in the cited text

the first time through i expect but’ve done 0–100 for sure

one way or another and indeed in many cases many times.

the rules as i’ve always taken ’em for granted *do* allow

the 4/9 variation you mentioned.

,

for the unlikely reader both

ignorant of such repeated-single-digit

decimals and interested to learn more,

is the result of doing a “long division”

for 4/9.

.333333… = 1/3

is actually pretty well known

and the result easily follows

by handwaving:

obviously one-third of *this* is

.111111… = 1/9 follows.

and so on for .2-bar,

.4-bar, and all the others.

note… i used to be a math teacher…

that if x = .111111…

we have 10x = 1.111111….

and so 9x = 1

(subtracting the first equation

from the second gives the third):

x = 1/9 as we claimed already.

math is the only stuff you can

*ever* get *this* right.

anyway with any consistency.

back on “four fours”.

the book i learned it from *mentioned*

the gamma function. i think.

anyway i soon learned that

GAMMA(n) = (n-1)!

for positive integer n

but it was a *long* time

before i had any idea what

any such silly thing could be for.

other than a cheat.

one builds up a table

(in the “get mad” phase).

what can i do with…

…*one* four?

…*two* fours?

if this comes in handy.

## kibrolv said

ruth carver’s four 4’s page.

## cut & paste | the livingston review said

[…] in media’s rays ) cross-products11/15/09 use the source luke permutations11/16/09 never get outta the boat four 4’s11/17/09 there is no frigate like a boat trees11/18/09 the chinese room more […]

## Vlorbik On Math Ed | the livingston review said

[…] 11/14/09 in media’s rays ) cross-products 11/15/09 use the source luke permutations 11/16/09 never get outta the boat four 4’s 11/17/09 there is no frigate like a boat trees 11/18/09 the chinese room more […]