In the 1950s, physicists George Gamow and Moritz Stern worked in the same seven-story building. Gamow, on the second floor, noticed that the first elevator to arrive at his office was most often going down. For Stern, on the sixth floor, the first elevator was most often going up. It was as if elves were manufacturing elevator cars in the middle of the building.
You can observe the same phenomenon in most tall buildings, and there are no elves involved. Do you see why it occurs?
The top figure, measuring 8 × 8, can be reassembled to form the bottom figure, measuring 5 × 13. Thus 64 = 65.
— A.C. Orr, Literary Digest, 1906
Given any pair of potatoes — even bizarre, Richard Nixon-shaped potatoes — it’s always possible to draw a loop on each so that the two loops are identical in three dimensions.
Do you see the simple, intuitive proof for this?
5 is one number.
2 and 3 are 5.
Therefore 2 and 3 are one number.
175 = 11 + 72 + 53
I show you three cards. One is white on both sides, one is black on both sides, and one is white on one side and black on the other. I shake them in a hat, remove one at random, and place it on a table. The side that’s face up is black. What’s the probability that the other side is also black?
Hint: It’s not 1/2.
In 1626 Peter Minuit, first governor of New Netherland, purchased Manhattan Island from the Indians for about $24. … Assume for simplicity a uniform rate of 7% from 1626 to the present, and suppose that the Indians had put their $24 at interest at that rate … and had added the interest to the principal yearly. What would be the amount now, after 280 years? 24 × (1.07)280 = more than 4,042,000,000. [The current value of Manhattan is] a little more than $4,898,400,000. … The Indians could have bought back most of the property now, with improvements; from which one might point the moral of saving money and putting it at interest!
— W.F. White, A Scrap-Book of Elementary Mathematics, 1908
Take any number and rearrange its digits to form another number.
Subtract one from the other. The difference will always be divisible by 9.
Schoolmaster: Suppose x is the number of sheep in the problem.
Pupil: But, sir, suppose x is not the number of sheep.
Mathematician J.E. Littlewood remarks: “I asked Prof. Wittgenstein was this not a profound philosophical joke, and he said it was.”
Lend me $10, but give me only half of it.
Then you’ll owe me $5, and I’ll owe you $5, and we’ll be even.
Nothing is heavier than lead.
Feathers are heavier than nothing.
Therefore feathers are heavier than lead.
“Ignorance more frequently begets confidence than does knowledge.” — Charles Darwin
Born in 1804, Zerah Colburn was thought to be mentally retarded until the age of 7, when his father overheard him solving multiplication problems for other children and discovered he was a prodigy. From the 1872 autobiography of Amos Kendall, with whom he boarded briefly:
He could multiply together any two numbers under a hundred in less than a minute. He could tell, apparently without thought, how many days there are in any number of years less than thirty, and in any number over thirty and up to a hundred upon a minute’s reflection. After being told the denominations of weights and measures, he would reduce one to another with the greatest readiness. He answered correctly the question, ‘How many gills are there in three barrels?’ The question, ‘How many are 25 × 25 + 35 × 35 +45 × 45?’ he answered correctly with little hesitation. He readily multiplied any number over a hundred by any number less. In less than a minute he answered correctly the question, ‘How many days are there in seventy-three years?’
“What rendered his performances more wonderful was, that he did not know a figure when written, and could not count more than fifty. How he knew the names of larger numbers was a mystery, and he was sometimes embarrassed in making his answers understood. After he had stated correctly the number of days in a given number of years, he was asked how many hours there were. He said he did not know the number of hours in a day. On being told it was twenty-four he immediately gave a correct answer.”
A man deposited $50 in a savings account, then withdrew it in various sums. When he’d recovered his $50 he was surprised to find $1 left in the account, though it had drawn no interest. When he inquired, the bank produced this ledger:
2 + 5 + 6 = 13; 132 = 169
1 + 6 + 9 = 16; 162 = 256
83 = 512; 5 + 1 + 2 = 8
273 = 19683; 1 + 9 + 6 + 8 + 3 = 27
Passing through the quadrangle of Christ Church, Oxford, one day, the classical scholar Gilbert Murray encountered Albert Einstein sitting dreamily in thought.
Murray asked him what he was thinking about.
“I am thinking,” Einstein answered, “that, after all, this is a very small star.”
He who has not lost a thing has it.
You have not lost horns.
Therefore you have horns.
Is this a bad sum?
Not in a mirror:
Adapted by Martin Gardner from Henry Dudeney.
9 × 9 + 7 = 88
98 × 9 + 6 = 888
987 × 9 + 5 = 8888
9876 × 9 + 4 = 88888
98765 × 9 + 3 = 888888
987654 × 9 + 2 = 8888888
9876543 × 9 + 1 = 88888888
98765432 × 9 + 0 = 888888888
You say that you have a dog.
Yes, and a villain of a one, said Ctesippus.
And he has puppies?
Yes, and they are very like himself.
And the dog is the father of them?
Yes, he said, I certainly saw him and the mother of the puppies come together.
And is he not yours?
To be sure he is.
Then he is a father, and he is yours; ergo he is your father, and the puppies are your brothers.
Let me ask you one little question more, said Dionysodorus, quickly interposing, in order that Ctesippus might not get in his word: You beat this dog?
Ctesippus said, laughing: Indeed I do; and I only wish that I could beat you instead of him.
Then you beat your father, he said.
— Plato, Euthydemus
“Anton Von Leewenhoek
Has a small problem,” con-
Fided his wife.
Doesn’t disturb me; his
Blighting my life!”
— Theodore L. Drachman