The Eighth Night
•
�
Robert was up at the blackboard. His two best
friends in the class-Al, the one he played football
with, ,rnd Betsy, the one with the pigtails-were
sitting in front, as usual. And, as usual, they were
having an argument.
Just what I needed, thought Robert. A dream
about school!
At that very moment the door flew open and in
came not Mr. Bock.cl but-the number devil.
"Good morning," he said." Arguing again, I sec."
"Betsy's sitting in my place," said Al.
"Then switch places."
"She won't budge."
"Put an A for Al and a B for Betsy on the board,
Robert," said the number devil.
Why not, Robert thought, if it makes him happy.
1-1- ')
"Now, Betsy," said the number deYil, "I want
you to change places with Al."
And for some strange reason Bets\· didn't make
a scene. She stood up and changed places with Al.
<
•
If thi� keeps up, Robert said to himself, we can
forget <1bout the clas".
And it did keep up, bccau�e Al decided he
wanted to sit on the left.
"Which means we clll have to get up," said Betsy.
"I don't know why, but ...Come on, Charlie."
Herc is how it looked when thev'd all settled
down ag,1in:
.!
wrote Robert on the board.
At that very moment the door flew open <1gain
and in came Charlie, late as usual. Clurlie �at
down next to Betsv.
wrote Robert.
But Betsy didn't like that. "If I'm going sit on
the left, I'm going to sit all the way on the left."
"Heavens to Betsy!" Charlie cried.
And the two changed place
Then Al was upset. "I want to sit next to Bctsv!"
he said, so easygoing Charlie gave Al his pl�ce
without a murmur:
,
,._. � .-
..
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; B A-C _;._j·:J
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/50
.
Not that it lasted, of course.
"I won't sit next to Charlie for another minute,"
said Betsy before long. She was a real pain, that
Bet�y. And since Betsy had to have her way, the
boys had to get up and move. Robert wrote:
"Enough is enough," he said. "That's it.''
''Is it, now?" the number devil responded. "The
three of them haven't exhausted the possibilities
open to them.Supposing Al were to sit on the left,
Charlie in the middle, and Betsy on the right?"
"Not on your life!" Betsy cried.
"Don't make such a fuss, Betsy!"
/5 J
The three of them pulled themselves up .1ga111
and repositioned themselves:
"Hey, Robert! Notice ,rnything? I don't think
those three will come up with anything el�e. ''
Robert looked at the board:
"\Xie seem to have tried all the possibilities," he
�aid.
"That we have," said the number de\·il. "B ut
there are more than the four of you in this cl.1s
s.
Quite ,1 few more, I'm afraid."
At th,1t very moment the door flew open and in
rc1n Doris, out of breath.
.
"Hcy, w I1at 's gom
g on here? Where's Mr.
Bockcl? And who arc vou?"
"I'm substituting for Mr. Bockel. He's taking the
dav off. He said he needed ,1 rest from the chaos
that reigns in this class."
"I can see wlw," said Doris. "Look at them.
Thev'rc all in the. wrong scats. Since when do you
sit ti1ere, Charlie? That's my desk!"
"What order do you suggest, Doris?" the
number devil asked.
"\Vhy not go by the alphabet?" she said. "A for
Al, B for Betsy, C for Charlie, .111d so on. That's the
simplest way out."
"As \'OU like. Let's give it a try."
So R.obert wrote the following on the board:
But the other� weren't the least bit happy with
Dori�\ suggestion, and all hell broke loose in
the cbssroom. Betsy was the worst, biting ,rnd
scr.uching when one of the others refused to give
way, but they all pushed .111d shoved. Then, little by
!in.le, the cr.azy game of musical chairs they were
playing began to seem like fun, and they switched
places so fast that Robert hc1d trouble keeping up
with them. In the end, though, he did manage to
vet
all their scatincrt, combinations on the board:
t,
I 53
It\ a good thing there arc lots of kids .1b�ent
today, thought Robert, or it would go on forever.
And at that very moment the door flew open
.rnd in came Enrique, Felice, Gary, I 1 ugh, Iris,
Jamil, and Karen.
"No! No!" Robert cried. "No! Please! Don't sit
down or I 'II go crazy!"
"All right," said the number devil. "That's it for
today. You can all go home."
"Mc too?" Robert asked.
"No, I need you to st,1y .1 while."
While his classmates rc1n out into the -;choolyard,
Robert looked over the numbers on the board.
"Well," said the number devil. "Wl1c1t do you
make of them?"
"I'm not sure," said Robert. "All I know is that
the number of ways they can sit increases awfully
fast. As long as there were only two kids, things
I 5./
".\'o.' \ o.'" Robert lned. "No.' Ple'1se.' D011 't :i1t dO'l,.)17 or I 'II go l!'aL) .'"
"All 11ght, "saul t!Jl m11nbcr devil. "That':, 11 (or today. You ca11 all go humc."
were simple: two kids, two possibilities. But with
three kids there were six possibilities. And with
four-wait a minutc-twcntv-four."
"What if there's only one?"
"What a thing to c1 k!Only one possibility.''
"Let's try multiplying," s.1id the number devil.
Cluldre11,;
'f'o55ibilitie5:
"Utter confusion. Pandemonium," replied the
number devil. "I can just see them pushing ,1nd
shouting, trying out each and every combination.
It would have taken ages.Together with Al, Betsy,
and Charlie there would have been eleven of chem.
That means eleven vroom! possibilities. Can you
guess how ma.ny that is?"
"I know I can't do it in my head, but I always
bring my calculator to school-I have to hide it, of
course: Mr. Bock.cl can't stand the sight of
calculators-so I'll have the answer for you in .1
jiffy.
"I sec, " saw
·JRobere."I ntcrcst111g.
. "
"The more of your classmates join in the g,1me,
the more inconvenient it is to write it out like chat.
There's .1 shorter way, though.You uke the num­
ber .111d put an exclamation mark ,1fcer it.Like chis:
And you read it: four vroom!"
"What do you chink would have happened if you
hadn't sent Enrique, Felice, Gary, Hugh, Iris,
Jamil, and Karen home?"
"Eleven vroom!" he said excitedly, "is precisely
39,916,800.Wow!Nearlv
., 40 million!"
"So if we'd gone through all the combination�,
we'd still be here eighty years from now. Your
classmates would be in wheelchairs and we'd have
to hire eleven nurses to do the pushing. See how
useful a bit of mathematics can be? Which remind�
me ... Have a look out of the window and tell me
if your classmates arc still there."
''Oh, I'm sure chev'vc all gone their separate
ways.
fir,
/P
-
"I assume vou
shake hands when \'OU
sa\·• o·bood.
.
bye.''
"Shake hands!\Y/e mumble, 'Sec vou'-it ,·ou're
lucky.' '
"A pity," said the number de,il, "because I
wonder how long it would take for each of them to
shake hands with e,1ch of the others."
"You know perfectly well there'd be an untold
number of handshakes. Eleven vroom! of them, I
"upposc, since there arc eleven of them."
"Wrong!'' said the number devil.
"\Y/ait ,1 minute," said Robert. "I sec. If there
were two of them, thcv'd need onh· one hand­
shake.If there were three ..."
"Tr: putting it on the bo,ud."
T his is what Robert wrote:
'People.
Robert couldn't remember, so the number devil
made ,1 few big dots on the bo ard:
"Coconuts!" Robert shouted. "Triangle num­
bers!"
"And how do they go?"
Robert wrote on the board:
Ha.K.dsk.axes:
"Two people-one handshake. Three people­
three handshakes. Four people-six handshakes.
Five people-ten."
"One, three, six, ten ... Look familiar?"
"So you d
' need exactly fifty-five handshakes."
"Tlut wouldn't be so bad," said Robert.
"And here\ what you do to get around all tlut
,uithmetic. You draw a few circles on the bc1c1 rd:
The letters stand for your friends: A for Al, B for
Betsy, C for Charlie, and so on.
"Then you join the letters with lines:
I t,C
"Pretty, isn't it? And since e.1ch line represenb a
l1c1ndsh,:tke, all you have to do is count them."
"One, three, six, ten, f ifteen ...As usual," said
Robert. ''There's only one thing I don't under­
stand. How is it that everything you do works?"
"That's the devilish thing about numbers: every­
thing works. \'v'ell, almost everything. Because the
prima-donna numbers-remember them?-they
have their problems. And you've got to keep your
eyes peeled for them or you'll fall flat on your face.
Which is why so many people hate numbers. I
can't "und slobs, and they can't stand numbers.
B: the w,1:·, go to the window and you'll sec a
"chooly,ud tlut looks like a pigsty."
Robert had to ,1dmit it. The schoolvard w,1s
strewn with soda cans, newspapers, and sandwich
wrappmgs.
"If three of you pick up some brooms, you c1n
sweep it clean in half ,111 hour."
"\'v'hich three have you got in mind?'' Robert
a"ked.
"Al, Betsy, and Charlie, say.Or Doris, I:nrique,
and Felice. And :'ou've got G,uy, Hugh, Iris,
Jamil, and Karen waiting in the wings."
"So which three doesn't matter."
"Right.,,
"Then we c.1n combine them ,111y old way," said
Robert.
It, I
l
"Right again. But "upposing the�' 're not all a,"<1il­
.1blc. Supposing Doris, Enrique, .rnd Felice aren't
avaibblc, so we lu\'c only three: Al, Beby, and
Charlie."
"Then thev'll have to do it."
"Good. Put tlut on the bo.ud."
And Robert ,vrotc:
"Now, if Doris runs in late as usual, what do we
do? What arc the possibilities?"
Robert thought .1 moment and wrote the
following:
"These four," he said.
"Now Enrique turns up. \Xlhy shouldn't he be
included? That makes five candidates. Sec what
vou can do with five."
But Robert refused. He was gcttmg a little
nervous.
"You tell me," he said.
"All right. With three people we can have only
one group of three; with four we can ha\'c four.
And with five we can have ten. Here, let me put it
on the board:
Jr,2
"There\ "omcthing spcci.11 I want you to notice.
As you can ..,cc, I've put the groups in alphabctic.11
order. How nuny groups begin with Al? Ten.
Four. And with Charlie
How manv.' with Bets\'?
.
only one. The same numbers keep coming up:
vuess how it goes on from there? I mean,
Can vou
t,
..
if we .1dd a few more 11ame...,: Felice, Gary, Hugh,
and so forth. How man�· groups would we have
then?"
"Be.ns me," said Robert.
"You remember how we cracked the handshake
I,_
/1,)
problem? W hen ever ybody shakes hands with
ever vbodv else?"
"That was a breeze. We used the triano-le
::,
numbers:
'
,I
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But they won't help with our broom brigades,
which work three to a g roup."
"Now what if .vou add the first two trianl�tilar
::,
numbe rs together?''
"That makes four."
"And the next one."
"That nukes ten."
"And the next."
"10 + l O = 20."
"Go on. ,,
"You mean keep going until I get to the
eleventh? You c.m't be serious."
"Don't worry. You can get there without aritl1 metic. You can ...�et
there without l�ucs
sino-t,, even
,
t,
withoutABCDEFGH!jf(."
"HOW!'"'"
"With our good old number tr iangle," said the
number de\ ii.
"You mean you're going to put one on the
board?"
"Heavens no! Not when I have mv walkinl� stick
.
b
handv."
Ir,./
No sooner did he tap the stick on the board than
there it was, in all its glory. In glorious color too.
"Couldn't be easier," he said. "For the hand­
shakes you count from top to bottom using the
green cubes: one handshake for two people, three
for three people, and fifty-five for eleven people.
"For our broom-brigade trio you use the red
cubes, again from top to bottom. We start out with
three people and one possibility. W hen we have
four people to choose from, we have four com­
binations. W ith f ive people we have ten. How
many would we have if all eleven of your class­
mates show up?"
"165," Robert answered. "You're right. It is easy.
This number triangle is nearly as good as a com­
puter. But tell me, what are the orange cubes for?"
"Oh, them," said the number devil. "Well, as
vou mav have noticed, we number devils aren't
easily satisfied; we tend to go overboard. So just in
case ·three people couldn't handle the clean-up and
you needed a fourth, I wanted you to know how
�nany possibilities you'd have. How many will
there be if, say, eight people apply for a broom­
brigade qua rtet?"
"SeYenty," said Robert, having no trouble find­
ing the answer in the number triangle.
"Correct," said the number devil. "By now you
can guess what the blue cubes are for."
,I
'
Ir, 'i
s.1id Robert. "It I
"The broom-brigc1de octet,''
only one p o -;sibility.
h.we eight \·olunteers, I haYc
. And so on."
But \\ itl1 ten I h.wc forty-fi\'c
«You get the picture."
ool\',Hd looks like
"\'v'lut do vou think the sch
t of the window. lt
now?" Robert s,1id, looking ou
seen it. •<I wonder
was cleaner than he had ever
which three did it?''
them, my dear
''\Vell, you weren't .1.mong
Robert.''
the schooly ard
''How do you expect me to sweep
er.., and cubes .1t me
when you keep throwing numb
lt?,,
o n ::,
.111 11 itrh
r-, t l
m me."
"M.1� be you need J. little rest fro
u corning b,1ek?"
"\'v'lut do you me.111? Aren't yo
holiday," ... aid the
''I thought I could u-;e ,1 little
ulk numbers with
number de\'il. "You c.1.n ,1\w.ws
Mr. Bockcl."
the world, but
It w.1sn't the gre.uest pro spect in
go bcKk to ..,chool
he Ju d no cho ice. And he Jud to
the ne;\t d.w. .1m. ·w,1\·. .
ro om, he saw Al
\'v'hen he w.1\ked into the cbss
. Neither ..,ecmcd
.1nd Bet'>)' in their nornul pbces
keen on S\Yitching.
ic called out.
'' Herc comes the wiz.ud! '' Ch,u-l
sleep!" Betsy
"Robert docs problems in his
te,1sed.
Ir,�
"Do you think it helps?" asked Doris.
"Not very much," said Jamil . "Mr. Bockcl can't
tand him."
"Well, the feeling's mutual," Robert replied.
Robert stoic a glance at the schoolyard.
As usual, he thought . A regular dump! So much
)r his dream. But the numbers remained. He
}tdd count on the numbers.
At that moment the door flew open and in
alkcd the inc\'itablc Mr. Bockcl with a briefcase
10ck-full of pretzels.
I r.s
The Ninth Night