A puzzle by A. Kozlov from the Soviet popular science magazine Kvant:
Watching a solar eclipse, a girl asks her father how much farther away is the sun than the moon.
He says, “As far as I remember, 387 times farther.”
She says, “Then I can figure out how much greater is the sun’s volume than the moon’s.”
He thinks about this and says, “I think maybe you can.” How can she do this?
A puzzle from Stuart Collingwood’s Lewis Carroll Picture Book, 1899:
A captive Queen and her son and daughter were shut up in the top room of a very high tower. Outside their window was a pulley with a rope round it, and a basket fastened at each end of the rope of equal weight. They managed to escape with the help of this and a weight they found in the room, quite safely. It would have been dangerous for any of them to come down if they weighed more than 15 lbs. more than the contents of the lower basket, for they would do so too quick, and they also managed not to weigh less either.
The one basket coming down would naturally of course draw the other up.
The Queen weighed 195 lbs., daughter 105, son 90, and the weight 75.
How did they do it?
In 1961, irate at receiving a bill for an £85 surtax from the Inland Revenue, A.P. Herbert sent them a check in verse:
Dear Bankers, PAY the undermentioned hounds
The shameful sum of FIVE-AND-EIGHTY POUNDS
By “hounds,” of course, by custom, one refers
To SPECIAL INCOME TAX COMMISSIONERS:
And these progenitors of woe and worry
You’ll find at LYNWOOD ROAD, THAMES DITTON, SURREY.
This is the second lot of tax, you know,
On money that I earned two years ago.
(The shark, they say, by no means nature’s knight,
Will rest contented with a single bite:
The barracuda, who’s a fish more fell,
Comes back and takes the other leg as well.)
Two years ago. But things have changed since then.
I’ve reached the age of threescore years and ten.
My earnings dwindle; and the kindly State
Gives me a tiny pension — with my mate.
You’d think the State would generously roar
“At least he shan’t pay surtax any more.”
Instead by this un-Christian attack
They get two-thirds of my poor pension back.
Oh, very well. No doubt it’s for the best;
At all events, pray do as I request;
And let the good old customs be enforced —
Don’t cash this check, unless it is endorsed.
To his astonishment he received this reply:
Dear Sir,
It is with pleasure that I thank
You for your letter and the order to your bank
To pay the sum of five and eighty pounds
To those here whom you designate as hounds.
Their appetite is satisfied. In fact,
You paid too much and I am forced to act,
Not to repay you, as perchance you dream,
Though such a course is easy, it would seem.
Your liability for later years
Is giving your accountants many tears;
And ’til such time as they and we can come
To amicable settlement on the sum
That represents your tax bill to the State
I’ll leave the overpayment to its fate.
I do not think this step will make you frown:
The sum involved is only half-a-crown.
Yours faithfully,
A.L. Grove
He wrote back:
I thank you, Sir, but am afraid
Of such a rival in my trade:
One never should encourage those —
In the future I shall pay in prose.
“Travelling is one Way of lengthening Life, at least in Appearance. It is but a Fortnight since we left London; but the Variety of Scenes we have gone through makes it seem equal to Six Months living in one Place.” — Benjamin Franklin, letter to Mary Stevenson, from Paris, Sept. 14, 1767
First-Sergeant Thomas Innes Woods, of Company B, was killed on May 8th [1864]. The first time that Sergeant Woods was ever known to ask permission to leave his post on march or in battle occurred this day, after the Regiment’s all-night march to reach Spottsylvania ahead of Lee. When it became evident that a battle was imminent, Sergeant Woods asked Captain H.W. Grubbs for a pass to go to the rear. On his declaring that he was not sick, he was advised by the Captain that under the circumstances he could not be excused, and Sergeant Woods resumed his post at the head of the Company. Shortly after, during a halt by the roadside, Sergeant Woods wrote in his diary the following, addressed to his friend, Sergeant James A. McMillen: ‘I am going to fall to-day. If you find my body, I desire you to bury it and mark my grave so that if my friends desire to take it home they can find it. Please read the Ninetieth Psalm at my burial.’ He was killed early in the battle. His body was found by Sergeant McMillen and others of Company B, the diary being found in his pocket. His request for the Ninetieth Psalm to be read at the grave was complied with.
— Charles F. McKenna, ed., Under the Maltese Cross, Antietam to Appomattox: The Loyal Uprising in Western Pennsylvania, 1861-1865: Campaigns 155th Pennsylvania Regiment, 1910
In Pale Fire, Nabokov notes an “absolutely extraordinary, unbelievably elegant” verbal curiosity:
A newspaper account of a Russian tsar’s coronation had, instead of korona (crown), the misprint vorona (crow), and when next day this was apologetically ‘corrected,’ it got misprinted a second time as korova (cow).
“The artistic correlation between the crown-crow-cow series and the Russian korona–vorona–korova series is something that would have, I am sure, enraptured my poet,” he wrote. “I have seen nothing like it on lexical playfields and the odds against the double coincidence defy computation.”
In 1943, fed up with modernist poetry, two Australian servicemen invented a fake poet and submitted a collection of deliberately senseless verses to a Melbourne arts magazine. To their delight, they were accepted and their author hailed as “one of the most remarkable and important poetic figures of this country.” In this week’s episode of the Futility Closet podcast we’ll tell the story of the Ern Malley hoax, its perpetrators, and its surprising legacy in Australian literature.
We’ll also hear a mechanized Radiohead and puzzle over a railroad standstill.
Canada’s “victory nickel,” struck from 1943 to 1945, included a special message to stimulate the war effort: Engraved around the rim were the words WE WIN WHEN WE WORK WILLINGLY in Morse Code.
The coin was reissued in 2005 to commemorate the 60th anniversary of V-E Day.
Princeton mathematician John Horton Conway investigated this curious permutation:
3n ↔ 2n
3n ± 1 ↔ 4n ± 1
It’s a simple set of rules for creating a sequence of numbers. In the words of University of Calgary mathematician Richard Guy, “Forwards: if it divides by 3, take off a third; if it doesn’t, add a third (to the nearest whole number). Backwards: if it’s even, add 50%; if it’s odd, take off a quarter.”
If we start with 1, we get a string of 1s: 1, 1, 1, 1, 1, …
If we start with 2 or 3 we get an alternating sequence: 2, 3, 2, 3, 2, 3, …
If we start with 4 we get a longer cycle that repeats: 4, 5, 7, 9, 6, 4, 5, 7, 9, 6, …
And if we start with 44 we get an even longer repeating cycle: 44, 59, 79, 105, 70, 93, 62, 83, 111, 74, 99, 66, 44, …
But, curiously, these four are the only loops that anyone has found — start with any other number and it appears you can build the sequence indefinitely in either direction without re-encountering the original number. Try starting with 8:
…, 72, 48, 32, 43, 57, 38, 51, 34, 45, 30, 20, 27, 18, 12, 8, 11, 15, 10, 13, 17, 23, 31, 41, 55, 73, 97, …
Paradoxically, the sequence climbs in both directions: Going forward we multiply by 2/3 a third of the time and by roughly 4/3 two-thirds of the time, so on average in three steps we’re multiplying by 32/27. Going backward we multiply by 3/2 half the time and by roughly 3/4 half the time, so on average in two steps we’re multiplying by 9/8. And every even number is preceded by a multiple of three — half the numbers are multiples of three!
What happens to these chains? Will the sequence above ever encounter another 8 and close up to form a loop? What about the sequences based on 14, 40, 64, 80, 82 … ? “Again,” writes Guy, “there are many more questions than answers.”
(Richard K. Guy, “What’s Left?”, Math Horizons 5:4 [April 1998], 5-7; and Richard K. Guy, Unsolved Problems in Number Theory, 2004.)
When Danish naval officer Gustav Holm was exploring the eastern coast of Greenland in 1885, an Inuit named Kunit gave him this three-dimensional wooden map.
The two parts form one whole: The bottom carving represents the coast from Sermiligak to Kangerdlugsuatsiak, and the top is an island offshore. The Inuit would carry these maps in their kayaks to navigate the waters between the two landmasses.
(From David Turnbull, Maps Are Territories, 1989.)