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English Pages [764] Year 2011
mm mm mm mm mm mm mm
SeleBed Papers on “Tun
OCT
2011
& Qames
CSLI Lecture Notes Number 192
(Q
«
Selected ‘Tapers on
‘Tun
&
Qames
«
fa Donald
£QP$; P&gQ; £^5
T V
E.
Knuth
CSLI Publications Stanford. California
Q
ep^
(p^5
(absolute)
Force
force which,
A
.
actin
when
causes for 1 kovac,
per of 1 potrzebie 1 1000000 bal
it
to
kova.
fu*
=
Work Energy and 1 i
blintzalpotrzebie
onnnoo h
=
1
=
1
bah ^b)
Power
_ 1 whatmeworry hah per kovac folowhatmeworry =
1
1000 100
WMW
KWMW

=
aeolipower (A.P.
1
Radioactivity The
quantity of
of 1 blintz
(WMW) (KWMW)
L
vhich
is
i
rad
1
hyme
in equilibrium
(how’s your
Temperature optimum tem
(S) — 100° Smurdley (27° C) eating halavah Tiprature for
m
and Measures
5
6
Fun and Games
Selected Papers on
Heat Energy The amount
of heat necessary to raise
blintz of halavah 1°
=
1000 v
1
S
=
1
1
vreeble (v)
large vreeble (V)
Counting
= =
48 things
49 things
1
MAD
1
baker’s
MAD
Angular Measure
= 1 zorch ('") = 1 zygo (§) = 1 circle or circumference
100 quircits ("") 100 zorch 100 zygo
('")
(§)
Miscellaneous Measurements speed of light
=
114441 Fp/kov
cowznofski
=
1.14441 x 10 12
1
light
1
vreeble
1
cosmo per square potrzebie
1
faraday
=
Fp
574.8 hah
=
3.1416 bumbles
=
1
122400 blobs/blintz equivalent weight electronech = 5.580 x 10“ 15 hoo
1
blintz molecular weight of a gas
density of 1
H2 O =
=
70.4
Kn
.3183 blintzes per ngogn (distilled water)
atmosphere pressure
=
1.4531 b/p 2
=
335.79 p of
Hg
=
3234.4 p per kov per kov downward force of 1 b = 3234.4 bal gravity
Basic Conversion Factors
= =
1
inch
1
mile
1
potrzebie
(abbreviated table)
11.222 potrzebies .71105 furshlugginer potrzebies
=
1
2.2633 millimeters = .089108 inches = 2.2633 kilometers = 1.4064 miles = 4.4182 potrzebies = .44182 furshlugginer potrzebies ngogn = 11.59455 cc = .012252 quarts = 2.3523 teaspoons blintz = 36.42538361 grams kiloblintz = 80.3042 pounds furshlugginer blintz = 36.425 metric tons
1
wood =144 minutes
1
kovac
1
year
1
furshlugginer potrzebie
1
centimeter
1
kilometer
1
1 1
= .864 seconds = 3.6524 cowznofskis
1
watt = 3.456482 whatmeworries horsepower = 2.58 kilowhatmeworries aeolipower = 38.797 horsepower
1
vreeble
1
1
=
34.330 calories
The Potrzebie System
225 LAFAYETTE STREET
•
of Weights
NEW YORK
and Measures
NEW YORK
12,
•
CANAL 51894
February 18 , 1957 Mr. Don Knuth Caldwell A re. 71j. 36 W. Milwaukee 16, Wise.
Dear Don, Surprise. Bet you thought we forgot all about you. Actually, we held off "Potrzebie System" for two issues pending the realignment of our talent. We then made the changes necessary, and... It's coming out in MAD #33.
Illustrated by Wood. Credited to you. So, enclosed is a cheek for $25.00 for the rights to print it in MAD, with the slight changes.
Also, enclosed is a copy of No. 4., which you said you didn't hare. No. 5 wo don't even have* I
Thanks again for letting us use your revolutionary measuring system. We're sure its inclusion in MAD will mean absolutely nothing to the world of soienoe.
A1
Feldstein^
Editor
7
Fun and Games
Selected Papers on
8
Addendum When
I
devised this system in 1955
I
had absolutely no idea what the
word
‘potrzebie’ signified, if anything. I knew it only as a strangelooking jumble of letters that had been appearing regularly in as
MAD
a staple element of
learned that
I
me 19
it is
random weirdness.* Shortly
a clipping from page
November
(
afterwards, in college,
a notuncommon word in Polish. Friends later sent
1A
of
The Milwaukee Journal dated Sunday,
19 1989, which began with the following banner headline:
‘Nasza ziemia znow w potrzebie’ Our homeland is again in need Lech Walesa)
—
The MAD true number of
typesetters accidentally dropped out a digit from the
millimeters per potrzebie; they printed it incorrectly as ‘2.263348517438173216473’. In a letter to the editor of the Case Tech in 1959, I pointed out that the correct number is easily memorized:
Just remember the poem “If by doubts you are much dismayed, this rhyme, a helping hand, was purveyed.” Then, assuming you can spell “purveyed,” just count the letters in each word and you get 2.2633484517438, correct to 14 places.
I’m pleased to have this opportunity to set the record straight, after more than 50 years. Readers who wonder where this fundamental constant arose should try to calculate the exact number of potrzebies in 9 inches, using the United States standard inch that was in force when I made the crucial initial measurements.
*
MAD
‘Potrzebie’ made its debut on the letters page of #10 (April 1954), reader wrote “Please tell me what in the world ‘Furshlugginer’ means.” Editor Kurtzman replied, “It means the same as Potrzebie.” One
when a
MAD
month later, Jack Davis included the word in his contribution to #11; Will Elder, Bernard Krigstein, and Wallace Wood followed suit in #12. Then #13 printed letters from four different readers, asking respectively for definitions of ‘furshlugginer’, ‘potrzebie’, ‘blintzes’,
swer: Perhaps
it
might
in question, such as,
They f
stuffed the furshlugginer blintzes with halavah?
taste like potrzebie!”
Between 1893 and 1959, the famous Mendenhall ter equals 39.37
inch
and ‘halavah’. Anusage of the words
be all explained by illustrating the
“Who
— which
is
US
inches, exactly.
precisely 2.54
cm
A
rule stipulated that
1
me
simpler standard, the international
— was adopted
in
America on
1
July 1959.
Chapter 2 Official
Tables of the Potrzebie System
[The notes below, dated “Jugular 2, 13” (4 April 1956), were privately distributed in mimeographed form to members of the Wisconsin Junior
Academy of Science in
1956.
They served
as the source from which
MAD
editors subsequently extracted the article in the previous chapter.]
The Potrzebie System
of Weights and Measures
This new system of measuring, which
is destined to become the measuring system of the future, has decided improvements over the other
systems
now
common
It is based upon measurements taken on Milwaukee Lutheran High School, MilwauMAD Magazine #26 was determined to be 2.26334 84517 43817 32164 73780 74941 60547 91419 5114 mm. This unit is the basis for the entire system, and is called one potrzebie of length. The potrzebie has also been standardized at 3515.3516 wave lengths of the red line in the spectrum of cadmium. A complete table of the Potrzebie System is given below.
in
6912 at the Physics
Lab
use.
of
kee, Wis.,
when
Length
(see ruler at right)
1
potrzebie
.000001 p 1000 fp =
= 1
the thickness of
= 1
thickness of
MAD
#26
farshimmelt potrzebie
(fp)
millipotrzebie (mp)
mp = 1 centipotrzebie (cp) 10 cp = 1 decipotrzebie (dp) 10 dp = 1 potrzebie (p) 10 p = 1 dekapotrzebie (Dp) 10 Dp = 1 hectopotrzebie (Hp) 10 Hp = 1 kilopotrzebie (Kp) 1000 Kp = 1 furshlugginer potrzebie 10
9
(Fp)
10
Selected Papers on
Fun and Games
Volume = 1 ngogn (n) = 1 farshimmelt ngogn (fn) = 1 millingogn (mtn) 10 mtn = 1 centingogn (cn) 10 cn = 1 decingogn (dn) 10 dn = 1 ngogn (n) 10 n = 1 dekangogn (Dn) 10 Dn = 1 hectongogn (Hn) 10 Hn = 1 kilongogn (Kn) 1000 Kn = 1 furshlugginer ngogn (Fn) 1
cubic dekapotrzebie
.000001 n
1000 fn
Mass
= 1 blintz (b) = 1 farshimmelt blintz (fb) = 1 milliblintz (mb) 10 mb = 1 centiblintz (cb) 10 cb = 1 deciblintz (db) 10 db = 1 blintz (b) 10 b = 1 dekablint.z (Db) 10 Db = 1 hectoblintz (Hb) 10 Hb = 1 kiloblintz (Kb) 1000 Kb = 1 furshlugginer blintz (Fb) 1
ngogn of halavah*
.000001 b
1000 fb
Time 1
average rotation of the earth on
= 1 wolverton = 1 severin (sev) 100 sev = 1 davis (dv) 100 dv = 1 wood (wd) 10 wd = 1 price (pr) 10 pr = 1 elder (el) 10 el = 1 kurtzman (k#) .00000001 pr
its
axis
=
1
price (pr)
(wl)
1000 wl
Date October
1,
calendar.
*
Halavah
1952
On is
is
the
the day
MAD was first published according to the old
new calendar
a form of pie, and
heat of .31416.
it
this
is
price
1
of
kurtzman
1.
Kurtzmans
has a specific gravity of 3.1416 and a specific
Official
Tables of the Potrzebie System
before this date are referred to as “Before
MAD
(K.M.).”
dar for each kurtzman contains 10 elders
follows:
Tales (Tal.)
1
6
2 Calculated (Cal.)
To
3
Force
A
The
calen
Humor (Hum.)
7 In (In)
(To)
8
4 Drive (Dri.) 5
11
(B.M.)” and kurtzmans
Madi named as
after this date are referred to as “ Kurtzmano
A
(A)
9 Jugular (Jug.)
You (You)
10 Vein (Vei.)
(absolute)
force which,
causes
=
it
when
acting
upon
1
blintz of
to attain a velocity of
1
mass
for 1 severin,
potrzebie per severin
blintzal (bal)
1
1000000 bal
=
furshlugginer blintzal (Fbal)
1
Energy and Work 1
blintzalpotrzebie
1000000 h
=
1
=
1
hoo
(h)
hah (hh)
Power 1
=
hah per severin
1000 100
WMW =
KWMW =
1
1
(WMW) (KWMW)
whatmeworry
kilowhatmeworry
1
aeolipower (A.P.)
Temperature 100° Smurdley (S) 0° S
=
= optimum temperature for eating halavah
(27° C)
absolute zero
Heat Energy The amount
= 1000 v
1
of heat necessary to raise one blintz of halavah 1° S
vreeble (v)
=
1
large vreeble (V)
Light Energy
The
intensity in a given direction of one square potrzebie of surface
body radiator at the temperature of freezing platinum (about 681° S), the surface being normal to the given direction,
of a black
=
1
cosmo
(cos)
12
Selected Papers on
The luminous
Fun and Games
produced
in a solid angle of one steradian by a uniform point source of one cosmo at the vertex of the angle
=
1
flux
shermlock (shr)
The quantity
of light that passes a cross section of a
in
one severin when the flux
=
1
beam
The illumination on a
surface
all
points of which are at a distance
of 1 kilopotrzebie from a uniform point source of one
=
1
=
1
of light
one shermlock
is
shermlockseverin (shrsev)
cosmo
kilopotrzebiecosmo, or potcosmo (pc) The average brightness of a surface emitting or reflecting light at the rate of one shermlock per square dekapotrzebie of actual surface
bumble (bum)
(The prefixes used
in mass,
volume, and length units are applied to
these units and those following to
Magnetism and
show
Electricity
a.
Magnetic Pole Strength. The strength of a pole that will repel an equal like pole at a distance of one potrzebie in a vacuum with a force of one blintzal
b.
Magnetic Field Intensity.
=
1
klek
The
intensity of a magnetic field
when a pole of one klek placed at the point acted upon by a force of one blintzal at a point
= c.
1
Magnetic Flux Density. at a point in a
= d.
1
is
The flux vacuum when the
schvester (sch), also called
e.
that
when
1
the flux density
is
one schvester
bentney (ben) bentneys
=
1
kilofurshlugginer bentney
(KFB)
Magnetomotive Force. The magnetomotive force producing the flux when one hoo of work is required to move a pole of one klek completely around a magnetic circuit against the field intensity
= f.
1 1 Vo
’
’
’
x
’
‘,
‘
’
’
‘,
‘
’
‘,
‘With a
’
’
’
’
’
‘,
L
l
(
,
’
x')
)
(4)
where IVi
=
‘chick’,
IV2
=
‘quack’,
W
4
—
W
3
‘oink’,
=
‘gobble’,
fV5
=
‘moo’, IV6
=
‘hee’,
(5)
and
W'k — 1
W
k
for
k
^
6;
W'G =
‘haw’.
(6)
Actually MacDonald’s priority has been disputed by some scholars; Peter Kennedy [8, p. 676] claims that “I Bought Myself a Cock” and similar farmyard songs are actually much older.
The Complexity
The song
of order
m
is
=
5o
Sm =
of Songs
35
defined by
e,
i? 2
(W" ) Vm
(7)
1,
where
W" =
W
‘chicks’,
2
W'l
The length n
=
Sm
of
= ‘ducks’, W3 = ‘turkeys’, = ‘pigs’, W5 = ‘cows’, Wq =
'
'
m + 180ra + 4(m/i +
30
2
4
b
while the length of the corresponding schema c
Here
Ik
= 
string x.
=
20
W
k
The
\
m + 211 +
+
‘donkeys’.
(5)
is
+ =
(li
W^ and
•
•
•
+ m + l
Wj{\,
)
(
where
l
m) +
F am )
(ai H
(9)
is
ai
+
•
•
•
+ flm).
(
10 )
denotes the length of
\x\

result follows at once,
if
we assume
that
Ik
—
A and dk
=a
for all large k.
Note that the
coefficient (20
+ A + a)/\/30+2A
assumes
its
minimum
value at
A
=
max(l, a
—
10)
(11)
when a is fixed. Therefore if MacDonald’s farm animals long names they should make slightly shorter noises.
ultimately have
Similar results were achieved by a French Canadian ornithologist, his song schema “Alouette” [2, 15], and at about the same time by a Tyrolean butcher whose schema [5] is popularly called “1st das nicht ein Schnitzelbank?” Several other cumulative songs have been collected by Peter Kennedy 8 including “The Mallard” with seventeen
who named
.
[
] ,
and “The Barley Mow” with eighteen. More recent compositions, like “There’s a Hole in the Bottom of the Sea” and “I Know an Old Lady Who Swallowed a Fly,” unfortunately have comparatively large verses
coefficients.
A
fundamental improvement was claimed
in
England
in 1824,
when
the true love of U. Jack gave to him a total of 12 ladies dancing, 22 lords aleaping, 30
drummers drumming, 36
pipers piping, 40 maids amilking,
42 swans aswimming, 42 geese alaying, 40 golden rings, 36 collie birds, 30 french hens, 22 turtle doves, and 12 partridges in pear trees during the twelve days of Christmas
m
2
/2
+ m/3
gifts in
m days,
[11, 12, 13].
when summed,
This amounted to
m 3/ +
so the complexity appeared
0 (y/n) however, another researcher soon pointed out 10 that his computation was based on n gifts rather than on n units of singing. The correct complexity of order y/n/ log n was finally established (see [7]). to be
[
]
Fun and Games
Selected Papers on
36
Thus the partridge in the pear tree gave an improvement of only 1/ \/Iog n but the importance of this discovery should not be underestimated, since it showed that the rr 5 barrier could be broken. The next big breakthrough was in fact obtained by generalizing the partridge schema in a remarkably simple way. It was J. W. Blatz of Milwaukee, Wiscon;
who
sin,
discovered a class of songs
first
known
“m
as
on the Wall”; her elegant construction 2 appears first major result in the theory.
Bottles of Beer
in the following
proof
of the
Theorem
There exist songs of complexity O(logn).
1.
Consider the schema
Proof.
Vk =Tk
BW 7
Tk B V ‘If
one of those bottles should happen to
Tk  1
B
fall,’
W‘.’
(12)
where
B—
W= Tk
and where space
is
is
required to define
Tk
.
more If
is
(...
x 2 y 2 Xiyix 0 yo) 2
more commonly
x\y,
.
called a “perfect shuffle”
see Eq. 7.1.3(76) in
difficult to describe via
[1].)
The
0 where j s \
x
~y —
(
>••> j\> jo,
selection
formulas, but
exactly s of the bits of y are equal to
1,
it
too
is
namely
then
x ji x j)2
xjsi
Of course x ~ 0 = 0. Notice that y ~ y — 2 s — 1. All numbers in INTERCAL are either short (16 bits) or long (32 bits), and they’re always nonnegative. Thus short numbers are always integers that are less than 65536; long numbers are always integers that are less than 4294967296. INTERCAL allows you to specify any short number by providing 65536 constants written respectively as #0, #1, It also allows you to use up to 65535 short variables, called .65535, which can take on any short number as .1, .2, a value; similarly it permits up to 65535 long variables, called :1, :2, 65535. There are 65535 short array variables ( ,1 through ,65535) and 65535 long array variables (;1 through 65535 ) too. Lotsa variables.
directly, .
.
.
#65535.
respectively
.
.
.
.
.
.
:
;
We’re ready now to look at a short (and very trivial) example of INTERCAL code. Line 01 of the program excerpt in Table 1 says that The remaining lines set the last ,16080 is an array with 15 elements. seven of those elements respectively to (0,6,41,136,322,999,999,999). Notice that each statement in Table 1 begins either with PLEASE, DO, or PLEASE DO.
These three
possibilities are interchangeable, except that the
TPK Table
1.
INTERCAL
in
Initializing part of a small array of short
01
PLEASE DO ,16080 )
(lp)"
+e k 
J
n
=
(n —
j
1
j0.4,
simple manipulation this relation
equivalent to
n > so the
(log 5)/ (log(l
number
+ 2/(m 
of elevators
2)))
=
^(m 
1)
+
0(l/m);
must be roughly proportional to the number
of floors!
The
following table shows the dependence of
n on m:
3
4
5
10
20
100
1000
2
3
4
8
16
80
804
Acknowledgments I
wish to thank Peter Weiner for some stimulating discussions of this I also wish to thank the referee for several suggestions
problem, and
that improved the presentation in this paper.
References [1]
George
Gamow
and Marvin Stern, PuzzleMath (New York: Viking
Press, 1958). [2]
Donald E. Knuth, Fundamental Algorithms Volume 1 of The Art of Computer Programming (Reading, Massachusetts: Addison,
Wesley, 1968).
Chapter 13
Fibonacci Multiplication [Originally published in Applied
A
Mathematics Letters
1 (1988),
curious binary operation on the nonnegative integers
is
5760.]
shown
to be
associative.
A
wellknown theorem of Zeckendorf [1,3,4] states that every nonnegaunique representation as a sum of Fibonacci numbers, stipulate that no two consecutive Fibonacci numbers occur in the sum. In other words we can uniquely write tive integer has a if
we
n
= Fkr +

+ Fk2
+F
kl
»
kr
,
•
•
>k >h»
•
2
»
j" means that k > j where the relation “k numbers are defined as usual by the recurrence
Fo
=
Fi
0,
=
Fk = F
1,
fc
_!
+F
fc
+
_2
2.
r
0,
>
0,
(1)
The Fibonacci
>
for k
2.
(2)
Given the Zeckendorf representations
m = Fj let
F
\
q
Fn
n
,
= Fkr +
f
Fkl
,
(3)
us define “circle multiplication” to be the following binary operation: r
q
Fj b +k c
(4)
6=1 c=l
In particular, Fj o
The purpose
Fk —
Fj +k
of this note
,
is
if
>
2
=
l
and k >2.
j to prove that circle multiplication satisfies
the associative law: (Z
o
m)
o
n
87
o
(m
o n)
(5)
Fun and Games
Selected Papers on
88
The proof is based on a variant of ordinary radix notation that Fibonacci numbers instead of powers. Let us write (
Then
ds
.
.
.
(d s ... dido )
dido)p
=
d s Fs
+
•
•
+ d\F\ +
•
doFo
a Zeckendorf representation
is
(6)
.
and only
if
uses
if
the
following three conditions hold: Zl.
Each
digit di
Z2.
Each
pair of adjacent digits satisfies d*+idj
Z3. d\
=
do
—
is
0 or
1.
=
0.
0.
For example, here are the Zeckendorf representations for the numbers
= = 3 = 4 = 5 = 6 = 7 = 8 =
(ioo) F
9
(1000) F
10
(10000)f
11
(10100) F
12= 13 =
Addition of (thereby adding
(
(
(100100) F
14
(101000) F
15
1000000) F
16
1 is 1
100000) F
We
easy in radixF:
(1000100) F (1001000) F (1010000) F (1010100) F
(10000000) F
= = =
(10000100) F (10001000) F (10010000) F
simply set di 0. Since rule (9) is no longer applicable,
1 for all
=
just described transforms (ds that satisfies Z1 and Z2.
The
Proof.
result
is
process applied to (d s
with
all d!i
into (d't
with
.
.
< d'3
.
(8) will
have d 3
—
.
.
.
>
i
0 and d
t
2.
2 for alii
2,
the carrying process
d 1 d 0 ) F into a sequence
vacuously true when
s
d 3 0 0) F inductively produces hence the sequence (d s d 3 d 2 dido) F .
.
(d't
1,
(d't
.
.
.
djd'0 ) F
the carrying .
.
d':i d'2 d'x
.
)
F
transformed d 2 + d'2 < 1 or if d'3 = 1, further carries lead to termination without changing d[. Otherwise we 1;
.
(d 2 +d'2 )d[do) F
0 and 2
rule (9) sets do
< d2 1
3.
If d!x
and only Os and
The addition procedure
.
is
If
0. Thus we have n = ( 1 0 0)j? with n zeros after
of n,
.
.
.
.
,
the rightmost
Lemma
1.
m+
Moreover,
m>
also define 0
=
oo.
and n > 4, the radixF addition n > min(m, n) — 2.
If
2.
We
4
rn
+n
is
clean.
Proof. Lemma 1 shows that radixF addition never reduces the number of trailing zeros by more than 2.
Thus,
it
for
is
m
o n has a natural radixF interpretation, completely analogous to ordinary binary multiplication.
Circle multiplication
because
example,
6 o 12
= =
(100100) F o (1010100) F
(10010000) F
+ +
(1001000000) F
(100100000000) F
(10)
Fun and Games
Selected Papers on
90
because we have j'2 = 5, notation of (3); the three for c
=
1, 2, 3.
—
j\
is
if
=
k2
4,
m F o
kc
and fci = 2 in the sums X^b=i Fj b + k .
easy to see that circle multiplica
it
monotonic:
l
increase the
left
l
m o n,
n
for
Lemma
3.
RadixF addition of the partial products of
Proof.
The
partial product
k r 3> Ay  1
Ay
mo Fk
we have,
.
has
is
clean.
Since
fc
successively,
>
fc r
_x
,
2
>
Ay_ 2
,
1
>
fci
1
1
Lemma
2; all
Theorem.
of these additions are spanking clean.
Let the Zeckendorf representations of
— m= n — ^
Then the
mo n
mo Fk = m + > k + 2.
m o Fkr +m o Fkr _ m o Fkr + to o Fkrl + mo Fkr _ mo Fkr H ... m o Ffe by
(11)
0.
every partial product increases.
Fi p
+
Fj q
\
Fk r
•
i
+
'
threefold circle product p
(lom)on =
+ Fi a + Fi b Fj 2 + Fjl + Fk2 + Fkl
•
•
1
m, and n be
,
.
the threefold
is
l,
,
sum
t
J2J2J2 F
ia +j b +k c

(
12 )
a= 1 6=1 c= 1
By Lemma 3, each partial product (/ o m) o Fk can be obtained Proof. by a clean addition of the partial products (/ o Fj) o Fk of l o to, shifted left k. Hence (lom)oFk = Fi a+ j b+k and the result follows YZ=i i by summing over k = Ay Ay
EL
.
.
,
Since the
sum
in (12)
must be associative
We
.
is
,
.
symmetric
in
l,
to,
and
n, the circle
product
as claimed in (5).
can extend the proof of
Lemma 3
without difficulty to show that
the three fold addition in (12) is clean. Hence we obtain a similar tfold sum for the tfold circle product of any t numbers.
Fibonacci Multiplication
Fk
The Fibonacci number
+
“golden ratio” (1
is
asymptotically
y/Z )/2; hence
FjoFk ~ VZFjFk, follows that the circle product
k cj)
/\/Z,
where
(j>
is
91 the
we have as
j,
k
—
>•
oo.
(13)
mon is approximately yfh mn
as 2.24mn and n are large. But 1 o n is closer to as 2.62 n; and 2 o n is approximately 3 n « 2.12(2n). This paper was inspired by recent work of Porta and Stolarsky [2] who made the remarkable discovery that the more complicated operation It
when
to
to * is
n
= mn +
[4>m\
[(/)n\
associative. Their “star product” satisfies to *
n
as
3.62 mn.
This research was supported in part by National Science Foundation grant CCR8610181, and in part by Office of Naval Research contract N0001487K0502.
References [1]
C. G. Lekkerkerker, “Voorstelling van natuurlijke getallen door een
som van [2]
[3]
E.
Simon Stevin 29
(1952), 190195.
Zeckendorf,
somme de [4]
getallen van Fibonacci,”
A. S. Fraenkel, H. Porta, and K. B. Stolarsky, “Some arithmetical semigroups,” Progress in Mathematics 85 (1990), 255264.
“Representation des nombres naturels par une de nombres de Fibonacci ou de nombres de Lucas,” Bulletin Royale des Sciences de Liege 41 (1972), 179182.
la Societe
“A generalized Fibonacci numeration,” Fibonacci
E. Zeckendorf,
Quarterly 10 (1972), 365372.
Addendum Zeckendorf’s representation can actually be traced back hundreds of where the anonymous author of Prakrta
years to medieval India,
Paingala
(c.
1320) presented algorithms for the equivalent problem of
finding the nth possible syllable
and TO’
rhythm
in Sanskrit poetry.
given total
number
Using
a long syllable, the possibilities are
for
1010, 10000, 10010, etc., preceded
‘0’
for a short
0, 10, 100,
1000,
by as many 0s as needed to make a
of beats.
Yuri Matijasevich wrote to
had devised the operation product in his paper
“Two
mon
me on
1
May
1990 to point out that he
already in 1967.
He
called
it
the indirect
reductions of Hilbert’s tenth problem” [Sem
inars in Mathematics, V. A. Steklov Mathematical Institute 8 (1970), 6874, which is a translation of his original Russian article in 3a nncKH
Haynnhix CeMHHapoB JIeimurpajiCKoro O'rpejienmi MaTeMarmecKoro IlHCTHTyra HMemi B. A. CreKsiOBti 8 (1968), 145158].
Fun and Games
Selected Papers on
92
known
Matijasevich had not
he sent
me
that his multiplication
associative;
is
the following elegant algebraic proof of associativity, to com
plement the algorithmic proof that representation
(1), let
Ffcr+1
n=

tained from n by shifting the bits d s
wellknown identity
had found:
I
.
.
n has the Zeckendorf \Fk2+x +Fkl+ i be the number obd 0 of (6) left one place. Then the .
Fk +i = Fk+1 F + F t
If
fe
F)_i
= Fk+1 F + Fk Fi+1  Fk F t
t
implies that
mon =
run
+ mn — mn.
= (lom)n+(lom)n—(lom)n — + m(ln + In — 2 In) + m(ln — In — In + In)
Hence (lom)on
(lorh)n\{lom)(ji—n)
is symmetric in l and n, must be equal to lo(mon). Indeed, if we also let n — n — n denote the operation of shifting n’s
id
n
so
it
Zeckendorf representation one place to the
mon —
lomon — Imn + lmn — Moreover, n
=

+
_(n
and n
1
right,
we
discover that
mn = mn + mn + mn
rhn —
=
;
lmn. _(n
+
1)/— 1 < a n multiplication.
is
the Zeckendorf representation of n
nfi+n, where (d s
—
(1
—
by
satisfied
dido)
—l/\
,
all lie in ,
.
.
}
is
the interval
closed under
a m a n — a mon gives a very nice way product and why it is associative; that is,
Thus the
understand the
tj>.
...
=
\Jh)j2
rule
to
.
+ n) = (m o n)$ + to q n
+
(mcf)
.
Generalizations of his results to other numbers 9 instead of
where 6 any socalled “Pisot number” (an algebraic integer > 1 whose conall lie inside the unit circle) have been found by P. J. Grabner, A. Petho, R. F. Tichy, and G. J. Woeginger, “Associativity of recurrence multiplication,” Applied Mathematics Letters 7 (1994), 8590. ,
is
jugates
Note
to hackers: If
x
=
(d s
.
.
did 0 ) 2 corresponds to the Zeckendorf
representation of n, the following bitwise operations will cleverly convert it
to the representation of
y users, backslashes and curly
T^X
braces are lacking too.
But hey, $720=6 $ in this state. Calculus and physics teachers can have $dx/dt$, etc. (All special characters are ignored when comparing two plates; thus 7206 and dxdt would be indistinguishable !
from those examples, and so would #7+2/06 and dx/dt?, both of which happen to be presently available subject to approval by the authorities.)
I wonder if a fallacious proposition like $120=6!$ would also be acceptable to the North Carolina censors. Probably it would, thereby setting back education in the Southeast. Even worse, I fear that somebody will ask for BADQMATH and be proud to display it. But let’s not be
Selected Papers on
134
pessimistic;
everybody
Fun and Games
North Carolina deserves applause
for leaping
way ahead
of
else.
Besides ordinary punctuation, a few special characters are also available. New Mexico has an enchanting Zia Sun symbol, which their website illustrates
shape of
with the example VANOITY.
its
empire, which
I
New York
a blob in the
offers
won’t illustrate here.
Since 1994, California has allowed a single delimiter to appear on vanity plates, taking the place of a letter or digit. There are four choices:
Drivers can use either a plus sign or one of the unique symbols {#,*,»}. The senior coeditorinchief of this journal could be CiULER D if he moved to California; Steve
rently
up
The
Smale could be tLE BODY. Both of these are curIVMATHS, and L*SPACE.
for grabs, as are X+IY.
(I mean blank spaces, not L*spaces) are too complicated to explain here. Suffice it to say that most states allow you to insert them in order to improve readability.
rules for spaces
Examples on the Road now at how some mathematicians and/or their friends have by meeting these constraints. Cathy Seeley, who was president of the National Council of Teachers of Mathematics (NCTM) from 2004 to 2006, likes her license plate so Let’s look
risen to the occasion
much
that she included
it
in
an illustrated lecture that
I
found on the
Internet:
a c%?msth
_
TEXAS
Hi
*
DOMATH and
can
d