# Vlorbik's Diner

## 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…
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
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.

1. ### Suesaid

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.

2. ### kibrolvsaid

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.

$.\bar4 := .4444444...$,
for the unlikely reader both
ignorant of such repeated-single-digit
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?
$\Gamma (4) = 6$
if this comes in handy.

3. ### kibrolvsaid

ruth carver’s four 4’s page.

4. ### cut & paste | the livingston reviewsaid

[…] 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 […]

5. ### Vlorbik On Math Ed | the livingston reviewsaid

[…] 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 […]