THE
PARADOX PARTY.
"Is not life itself a paradox?"
C.L. DODGSON, Pillow Problems.
"It is a wonderful age!" said Mr. Allgood, and everybody at the table turned
towards him and assumed an attitude of expectancy.
This was an ordinary Christmas dinner of the Allgood family, with a
sprinkling of local friends. Nobody would have supposed that the above remark
would lead, as it did, to a succession of curious puzzles and paradoxes, to
which every member of the party contributed something of interest. The little
symposium was quite unpremeditated, so we must not be too critical respecting a
few of the posers that were forthcoming. The varied character of the
contributions is just what we would expect on such an occasion, for it was a
gathering not of expert mathematicians and logicians, but of quite ordinary
folk.
"It is a wonderful age!" repeated Mr. Allgood. "A man has just designed a
square house in such a cunning manner that all the windows on the four sides
have a south aspect."
"That would appeal to me," said Mrs. Allgood, "for I cannot endure a room
with a north aspect."
"I cannot conceive how it is done," Uncle John confessed. "I suppose he puts
bay windows on the east and west sides; but how on earth can be contrive to look
south from the north side? Does he use mirrors, or something of that kind?"
"No," replied Mr. Allgood, "nothing of the sort. All the windows are flush
with the walls, and yet you get a southerly prospect from every one of them. You
see, there is no real difficulty in designing the house if you select the proper
spot for its erection. Now, this house is designed for a gentleman who proposes
to build it exactly at the North Pole. If you think a moment you will realize
that when you stand at the North Pole it is impossible, no matter which way you
may turn, to look elsewhere than due south! There are no such directions as
north, east, or west when you are exactly at the North Pole. Everything is due
south!"
"I am afraid, mother," said her son George, after the laughter had subsided,
"that, however much you might like the aspect, the situation would be a little
too bracing for you."
"Ah, well!" she replied. "Your Uncle John fell also into the trap. I am no
good at catches and puzzles. I suppose I haven't the right sort of brain.
Perhaps some one will explain this to me. Only last week I remarked to my
hairdresser that it had been said that there are more persons in the world than
any one of them has hairs on his head. He replied, 'Then it follows, madam, that
two persons, at least, must have exactly the same number of hairs on their
heads.' If this is a fact, I confess I cannot see it."
"How do the bald-headed affect the question?" asked Uncle John.
"If there are such persons in existence," replied Mrs. Allgood, "who haven't
a solitary hair on their heads discoverable under a magnifying-glass, we will
leave them out of the question. Still, I don't see how you are to prove that at
least two persons have exactly the same number to a hair."
"I think I can make it clear," said Mr. Filkins, who had dropped in for the
evening. "Assume the population of the world to be only one million. Any number
will do as well as another. Then your statement was to the effect that no person
has more than nine hundred and ninety-nine thousand nine hundred and ninety-nine
hairs on his head. Is that so?"
"Let me think," said Mrs. Allgood. "Yes—yes—that is correct."
"Very well, then. As there are only nine hundred and ninety-nine thousand
nine hundred and ninety-nine different ways of bearing hair, it is clear
that the millionth person must repeat one of those ways. Do you see?"
"Yes; I see that—at least I think I see it."
"Therefore two persons at least must have the same number of hairs on their
heads; and as the number of people on the earth so greatly exceeds the number of
hairs on any one person's head, there must, of course, be an immense number of
these repetitions."
"But, Mr. Filkins," said little Willie Allgood, "why could not the millionth
man have, say, ten thousand hairs and a half?"
"That is mere hair-splitting, Willie, and does not come into the
question."
"Here is a curious paradox," said George. "If a thousand soldiers are drawn
up in battle array on a plane"—they understood him to mean "plain"—"only one man
will stand upright."
Nobody could see why. But George explained that, according to Euclid, a plane
can touch a sphere only at one point, and that person only who stands at that
point, with respect to the centre of the earth, will stand upright.
"In the same way," he remarked, "if a billiard-table were quite level—that
is, a perfect plane—the balls ought to roll to the centre."
Though he tried to explain this by placing a visiting-card on an orange and
expounding the law of gravitation, Mrs. Allgood declined to accept the
statement. She could not see that the top of a true billiard-table must,
theoretically, be spherical, just like a portion of the orange-peel that George
cut out. Of course, the table is so small in proportion to the surface of the
earth that the curvature is not appreciable, but it is nevertheless true in
theory. A surface that we call level is not the same as our idea of a true
geometrical plane.
"Uncle John," broke in Willie Allgood, "there is a certain island situated
between England and France, and yet that island is farther from France than
England is. What is the island?"
"That seems absurd, my boy; because if I place this tumbler, to represent the
island, between these two plates, it seems impossible that the tumbler can be
farther from either of the plates than they are from each other."
"But isn't Guernsey between England and France?" asked Willie.
"Yes, certainly."
"Well, then, I think you will find, uncle, that Guernsey is about twenty-six
miles from France, and England is only twenty-one miles from France, between
Calais and Dover."
"My mathematical master," said George, "has been trying to induce me to
accept the axiom that 'if equals be multiplied by equals the products are
equal.'"
"It is self-evident," pointed out Mr. Filkins. "For example, if 3 feet equal
1 yard, then twice 3 feet will equal 2 yards. Do you see?"
"But, Mr. Filkins," asked George, "is this tumbler half full of water equal
to a similar glass half empty?"
"Certainly, George."
"Then it follows from the axiom that a glass full must equal a glass empty.
Is that correct?"
"No, clearly not. I never thought of it in that light."
"Perhaps," suggested Mr. Allgood, "the rule does not apply to liquids."
"Just what I was thinking, Allgood. It would seem that we must make an
exception in the case of liquids."
"But it would be awkward," said George, with a smile, "if we also had to
except the case of solids. For instance, let us take the solid earth. One mile
square equals one square mile. Therefore two miles square must equal two square
miles. Is this so?"
"Well, let me see! No, of course not," Mr. Filkins replied, "because two
miles square is four square miles."
"Then," said George, "if the axiom is not true in these cases, when is it
true?"
Mr. Filkins promised to look into the matter, and perhaps the reader will
also like to give it consideration at leisure.
"Look here, George," said his cousin Reginald Woolley: "by what fractional
part does four-fourths exceed three-fourths?"
"By one-fourth!" shouted everybody at once.
"Try another one," George suggested.
"With pleasure, when you have answered that one correctly," was Reginald's
reply.
"Do you mean to say that it isn't one-fourth?"
"Certainly I do."
Several members of the company failed to see that the correct answer is
"one-third," although Reginald tried to explain that three of anything, if
increased by one-third, becomes four.
"Uncle John, how do you pronounce 't-o-o'?" asked Willie.
"'Too," my boy."
"And how do you pronounce 't-w-o'?"
"That is also 'too.'"
"Then how
do you pronounce the second day of the week?"
"Well, that I should pronounce 'Tuesday,' not 'Toosday.'"
"Would you really? I should pronounce it 'Monday.'"
"If you go on like this, Willie," said Uncle John, with mock severity, "you
will soon be without a friend in the world."
"Can any of you write down quickly in figures 'twelve thousand twelve hundred
and twelve pounds'?" asked Mr. Allgood.
His eldest daughter, Miss Mildred, was the only person who happened to have a
pencil at hand.
"It can't be done," she declared, after making an attempt on the white
table-cloth; but Mr. Allgood showed her that it should be written,
"£13,212."
"Now it is my turn," said Mildred. "I have been waiting to ask you all a
question. In the Massacre of the Innocents under Herod, a number of poor little
children were buried in the sand with only their feet sticking out. How might
you distinguish the boys from the girls?"
"I suppose," said Mrs. Allgood, "it is a conundrum—something to do with their
poor little 'souls.'"
But after everybody had given it up, Mildred reminded the company that only
boys were put to death.
"Once upon a time," began George, "Achilles had a race with a tortoise—"
"Stop, George!" interposed Mr. Allgood. "We won't have that one. I knew two
men in my youth who were once the best of friends, but they quarrelled over that
infernal thing of Zeno's, and they never spoke to one another again for the rest
of their lives. I draw the line at that, and the other stupid thing by Zeno
about the flying arrow. I don't believe anybody understands them, because I
could never do so myself."
"Oh, very well, then, father. Here is another. The Post-Office people were
about to erect a line of telegraph-posts over a high hill from Turmitville to
Wurzleton; but as it was found that a railway company was making a deep level
cutting in the same direction, they arranged to put up the posts beside the
line. Now, the posts were to be a hundred yards apart, the length of the road
over the hill being five miles, and the length of the level cutting only four
and a half miles. How many posts did they save by erecting them on the
level?"
"That is a very simple matter of calculation," said Mr. Filkins. "Find how
many times one hundred yards will go in five miles, and how many times in four
and a half miles. Then deduct one from the other, and you have the number of
posts saved by the shorter route."
"Quite right," confirmed Mr. Allgood. "Nothing could be easier."
"That is just what the Post-Office people said," replied George, "but it is
quite wrong. If you look at this sketch that I have just made, you will see that
there is no difference whatever. If the posts are a hundred yards apart, just
the same number will be required on the level as over the surface of the
hill."
"Surely you must be wrong, George," said Mrs. Allgood, "for if the posts are
a hundred yards apart and it is half a mile farther over the hill, you have to
put up posts on that extra half-mile."
"Look at the diagram, mother. You will see that the distance from post to
post is not the distance from base to base measured along the ground. I am just
the same distance from you if I stand on this spot on the carpet or stand
immediately above it on the chair."
But Mrs. Allgood was not convinced.
Mr. Smoothly, the curate, at the end of the table, said at this point that he
had a little question to ask.
"Suppose the earth were a perfect sphere with a smooth surface, and a girdle
of steel were placed round the Equator so that it touched at every point."
"'I'll put a girdle round about the earth in forty minutes,'" muttered
George, quoting the words of Puck in A Midsummer Night's Dream.
"Now, if six yards were added to the length of the girdle, what would then be
the distance between the girdle and the earth, supposing that distance to be
equal all round?"
"In such a great length," said Mr. Allgood, "I do not suppose the distance
would be worth mentioning."
"What do you say, George?" asked Mr. Smoothly.
"Well, without calculating I should imagine it would be a very minute
fraction of an inch."
Reginald and Mr. Filkins were of the same opinion.
"I think it will surprise you all," said the curate, "to learn that those
extra six yards would make the distance from the earth all round the girdle very
nearly a yard!"
"Very nearly a yard!" everybody exclaimed, with astonishment; but Mr.
Smoothly was quite correct. The increase is independent of the original length
of the girdle, which may be round the earth or round an orange; in any case the
additional six yards will give a distance of nearly a yard all round. This is
apt to surprise the non-mathematical mind.
"Did you hear the story of the extraordinary precocity of Mrs. Perkins's baby
that died last week?" asked Mrs. Allgood. "It was only three months old, and
lying at the point of death, when the grief-stricken mother asked the doctor if
nothing could save it. 'Absolutely nothing!' said the doctor. Then the infant
looked up pitifully into its mother's face and said—absolutely nothing!"
"Impossible!" insisted Mildred. "And only three months old!"
"There have
been extraordinary cases of infantile precocity," said Mr. Filkins, "the truth
of which has often been carefully attested. But are you sure this really
happened, Mrs. Allgood?"
"Positive," replied the lady. "But do you really think it astonishing that a
child of three months should say absolutely nothing? What would you expect it to
say?"
"Speaking of death," said Mr. Smoothly, solemnly, "I knew two men, father and
son, who died in the same battle during the South African War. They were both
named Andrew Johnson and buried side by side, but there was some difficulty in
distinguishing them on the headstones. What would you have done?"
"Quite simple," said Mr. Allgood. "They should have described one as 'Andrew
Johnson, Senior,' and the other as 'Andrew Johnson, Junior.'"
"But I forgot to tell you that the father died first."
"What difference can that make?"
"Well, you see, they wanted to be absolutely exact, and that was the
difficulty."
"But I don't see any difficulty," said Mr. Allgood, nor could anybody
else.
"Well," explained Mr. Smoothly, "it is like this. If the father died first,
the son was then no longer 'Junior.' Is that so?"
"To be strictly exact, yes."
"That is just what they wanted—to be strictly exact. Now, if he was no longer
'Junior,' then he did not die 'Junior." Consequently it must be incorrect so to
describe him on the headstone. Do you see the point?"
"Here is a rather curious thing," said Mr. Filkins, "that I have just
remembered. A man wrote to me the other day that he had recently discovered two
old coins while digging in his garden. One was dated '51 B.C.,' and the other
one marked 'George I.' How do I know that he was not writing the truth?"
"Perhaps you know the man to be addicted to lying," said Reginald.
"But that would be no proof that he was not telling the truth in this
instance."
"Perhaps," suggested Mildred, "you know that there were no coins made at
those dates.
"On the contrary, they were made at both periods."
"Were they silver or copper coins?" asked Willie.
"My friend did not state, and I really cannot see, Willie, that it makes any
difference."
"I see it!" shouted Reginald. "The letters 'B.C.' would never be used on a
coin made before the birth of Christ. They never anticipated the event in that
way. The letters were only adopted later to denote dates previous to those which
we call 'A.D.' That is very good; but I cannot see why the other statement could
not be correct."
"Reginald is quite right," said Mr. Filkins, "about the first coin. The
second one could not exist, because the first George would never be described in
his lifetime as 'George I.'"
"Why not?" asked Mrs. Allgood. "He was George I."
"Yes; but they would not know it until there was a George II."
"Then there was no George II. until George III. came to the throne?"
"That does not follow. The second George becomes 'George II.' on account of
there having been a 'George I.'"
"Then the first George was 'George I.' on account of there having been no
king of that name before him."
"Don't you see, mother," said George Allgood, "we did not call Queen Victoria
'Victoria I.;' but if there is ever a 'Victoria II.,' then she will be known
that way."
"But there have been several Georges, and therefore he was 'George I.'
There haven't been several Victorias, so the two cases are not
similar."
They gave up the attempt to convince Mrs. Allgood, but the reader will, of
course, see the point clearly.
"Here is a question," said Mildred Allgood, "that I should like some of you
to settle for me. I am accustomed to buy from our greengrocer bundles of
asparagus, each 12 inches in circumference. I always put a tape measure round
them to make sure I am getting the full quantity. The other day the man had no
large bundles in stock, but handed me instead two small ones, each 6 inches in
circumference. 'That is the same thing,' I said, 'and, of course, the price will
be the same;' but he insisted that the two bundles together contained more than
the large one, and charged me a few pence extra. Now, what I want to know is,
which of us was correct? Would the two small bundles contain the same quantity
as the large one? Or would they contain more?"
"That is the ancient puzzle," said Reginald, laughing, "of the sack of corn
that Sempronius borrowed from Caius, which your greengrocer, perhaps, had been
reading about somewhere. He caught you beautifully."
"Then they were equal?"
"On the contrary, you were both wrong, and you were badly cheated. You only
got half the quantity that would have been contained in a large bundle, and
therefore ought to have been charged half the original price, instead of
more."
Yes, it was a bad swindle, undoubtedly. A circle with a circumference half
that of another must have its area a quarter that of the other. Therefore the
two small bundles contained together only half as much asparagus as a large
one.
"Mr. Filkins, can you answer this?" asked Willie. "There is a man in the next
village who eats two eggs for breakfast every morning."
"Nothing very extraordinary in that," George broke in. "If you told us that
the two eggs ate the man it would be interesting."
"Don't interrupt the boy, George," said his mother.
"Well," Willie continued, "this man neither buys, borrows, barters, begs,
steals, nor finds the eggs. He doesn't keep hens, and the eggs are not given to
him. How does he get the eggs?"
"Does he take them in exchange for something else?" asked Mildred.
"That would be bartering them," Willie replied.
"Perhaps some friend sends them to him," suggested Mrs. Allgood.
"I said that they were not given to him."
"I know," said George, with confidence. "A strange hen comes into his place
and lays them."
"But that would be finding them, wouldn't it?"
"Does he hire them?" asked Reginald.
"If so, he could not return them after they were eaten, so that would be
stealing them."
"Perhaps it is a pun on the word 'lay,'" Mr. Filkins said. "Does he lay them
on the table?"
"He would have to get them first, wouldn't he? The question was, How does he
get them?"
"Give it up!" said everybody. Then little Willie crept round to the
protection of his mother, for George was apt to be rough on such occasions.
"The man keeps ducks!" he cried, "and his servant collects the eggs every
morning."
"But you said he doesn't keep birds!" George protested.
"I didn't, did I, Mr. Filkins? I said he doesn't keep hens."
"But he finds them," said Reginald.
"No; I said his servant finds them."
"Well, then," Mildred interposed, "his servant gives them to him."
"You cannot give a man his own property, can you?"
All agreed that Willie's answer was quite satisfactory. Then Uncle John
produced a little fallacy that "brought the proceedings to a close," as the
newspapers say.
413.—A
CHESSBOARD FALLACY.
"Here is a diagram of a chessboard," he said. "You see there are sixty-four
squares—eight by eight. Now I draw a straight line from the top left-hand
corner, where the first and second squares meet, to the bottom right-hand
corner. I cut along this line with the scissors, slide up the piece that I have
marked B, and then clip off the little corner C by a cut along the first upright
line. This little piece will exactly fit into its place at the top, and we now
have an oblong with seven squares on one side and nine squares on the other.
There are, therefore, now only sixty-three squares, because seven multiplied by
nine makes sixty-three. Where on earth does that lost square go to? I have tried
over and over again to catch the little beggar, but he always eludes me. For the
life of me I cannot discover where he hides himself."
"It seems to be like the other old chessboard fallacy, and perhaps the
explanation is the same," said Reginald—"that the pieces do not exactly
fit."
"But they do fit," said Uncle John. "Try it, and you will see."
Later in the evening Reginald and George, were seen in a corner with their
heads together, trying to catch that elusive little square, and it is only fair
to record that before they retired for the night they succeeded in securing
their prey, though some others of the company failed to see it when captured.
Can the reader solve the little mystery?
Solution
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