A memorable puzzle from the Russian science magazine Kvant:

How can a goat, a head of cabbage, two wolves, and a dog be transported across a river if it’s known that the wolf is ‘culinarily partial to’ goat and dog, the dog is ‘on bad terms with’ the goat, and the goat is ‘not indifferent to’ cabbage? There are only three seats in your boat, so you can take only two passengers — animal or vegetable — at a time.

(You can keep order within the boat.)

Click for Answer

A Guest Appearance

enigma 11

Edward Elgar dedicated the eleventh of his Enigma Variations to George Robertson Sinclair, the organist of Hereford Cathedral.

“The variation, however, has nothing to do with organs or cathedrals, or, except remotely, with G.R.S.,” Elgar wrote. “The first few bars were suggested by his great bulldog, Dan (a well-known character) falling down the steep bank into the River Wye (bar 1); his paddling upstream to find a landing place (bars 2 and 3); and his rejoicing bark on landing (second half of bar 5). G.R.S. said, ‘Set that to music’. I did; here it is.”

After the river incident, Elgar had told a friend, “You wait till we get home. Japes!” He even marked bar 5 “Dan” in an early sketch of the piece. This would not have surprised Sinclair: Elgar had been in the habit of jotting down musical ideas, which he called “the moods of Dan,” in the organist’s visitor’s book, and sometimes these would find their way into later compositions. What Dan thought of all this is unrecorded.


Graham Greene once entered a magazine competition to parody the style of an author named Green(e). He parodied his own style and came in second. His entry, “The Stranger’s Hand,” was made under the pseudonym M. Wilkinson in the New Statesman‘s Week-end Competition No. 999 in 1949:

The child had an air of taking everything in and giving nothing away. At the Rome airport he was led across the tarmac by his aunt, but he seemed to hear nothing of her advice to himself or of the information she produced for the air hostess. He was too busy with his eyes: the hangars had his attention, every plane on the field except his own — that could wait.

‘My nephew,’ she was saying, ‘yes, that’s him on the list. Roger Court. You will look after him, won’t you? He’s never been quite on his own before,’ but when she made that statement the child’s eyes moved back plane by plane with what looked like contempt, back to the large breasts and the fat legs and the over-responsible mouth: how could she have known, he might have been thinking, when I am alone, how often I am alone?

Remarkably, he pulled the same coup in April 1961 (“I’m sorry but I’ve done it again”) with a fragment of autobiography set in verse; in August 1965 with a parodied biography of Sir Hugh Greene (his brother); and in April 1980 with “an extract from an imaginary novel by Graham Greene.” All but the last won prizes for successfully aping his own style. Not one to let good work go to waste, he developed two of these into legitimate projects, the 1949 entry into a script for a 1954 film and the 1980 effort into the opening of The Captain and the Enemy (1988).

(From Christopher Hawtree, ed., Yours Etc.: Letters to the Press, 1991.)

Plausible Deniability


The Boston Gazette published this cleverly seditious “Enigmatical Ballad” on June 24, 1782. If each line is read in full, the poem supports British rule of the American colonies, but if either the italic or the roman text is read alone, then it advocates revolution:

I justify every part, of King and Parliament,
Of a whig with all my heart, I hate their cursed intent;
For to support I’ll try, friends of administration,
Friends of Liberty, are troubles to the nation;
I think the association, a cruel, base intent,
An honor to the nation, the act of Parliament,
I wish the best success, to North and his conclusion,
Unto the grand Congress, the worst of all confusion;
All luck beneath the sun, to Mansfield, Bute and North;
To General Washington, destruction and so forth,
Hark! Hark! the trumpet sounds, the din of war’s alarms,
O’er seas and solid ground, doth call us all to arms;
Who for King George doth stand, their honors soon will shine;
Their ruin is at hand, who with the Congress join;
The acts of Parliament, in them I much delight;
I hate their cursed intent, who for the Congress fight,
The Tories of the day, they are my daily toast;
They soon will sneak away, who independence boast,
Who non-resistance hold, they have my hand and heart;
May they for slaves be sold, who act a whiggish part;
On Mansfield, North and Bute, may daily blessings pour,
Confusion and dispute, on Congress evermore;
To North, that British Lord, may honors still be done;
I wish a block or cord, to General Washington.

Podcast Episode 360: Haggard’s Dream


In 1904, adventure novelist H. Rider Haggard awoke from a dream with the conviction that his daughter’s dog was dying. He dismissed the impression as a nightmare, but the events that followed seemed to give it a grim significance. In this week’s episode of the Futility Closet podcast we’ll describe Haggard’s strange experience, which briefly made headlines around the world.

We’ll also consider Alexa’s expectations and puzzle over a college’s name change.

See full show notes …

Turning Point

Image: Wikimedia Commons

This pretty proof of the Pythagorean theorem is attributed to Leonardo da Vinci. Draw a right triangle and construct a square on each side, and make a copy of the original triangle and add it to the bottom of the hypotenuse square as shown. Now the shaded hexagon in the first figure can be rotated 90 degrees clockwise around the indicated point to occupy the position shown in the second figure. The orange and green quadrilaterals in the second figure are seen to be congruent to those in the first figure: The three shortest sides of the orange quadrilateral in the second figure correspond to their counterparts in the first, and the angles between them are assembled from the same constituents. The same is true of the green quadrilaterals. In each figure the shaded hexagon contains two instances of the original right triangle; remove these and we can see that the two squares in the first figure equal the large square in the second figure, proving Pythagoras.

10/10/2021 UPDATE: A number of readers point out that only the orange quadrilateral here can properly be said to turn; in the second diagram the green quadrilateral has been reflected as well. (Thanks, Mark and Bill.)

House Rules


From a letter from Mark Twain to Mabel Larkin Patterson of Chicago, Oct. 2, 1908:

The contents of your letter are very pleasant and very welcome, and I thank you for them, sincerely. If I can find a photograph of my ‘Tammany’ and her kittens, I will enclose it in this. One of them likes to be crammed into a corner-pocket of the billiard table — which he fits as snugly as does a finger in a glove and then he watches the game (and obstructs it) by the hour, and spoils many a shot by putting out his paw and changing the direction of a passing ball. Whenever a ball is in his arms, or so close to him that it cannot be played upon without risk of hurting him, the player is privileged to remove it to any one of 3 spots that chances to be vacant.

At the time his cats were named Apollinaris, Beelzebub, Blatherskite, Buffalo Bill, Sour Mash, Tammany, and Zoroaster — “names given them not in an unfriendly spirit,” he wrote, “but merely to practice the children in large and difficult styles of pronunciation.”

“It was a very happy idea. I mean, for the children.”


In a December 1985 letter to the Mathematical Gazette, Middlesex Polytechnic mathematician Ivor Grattan-Guinness writes that Astronomer Royal George Biddell Airy “would sometimes go around the Observatory, and on finding an empty box, insert a piece of paper saying ‘Empty box’ and thereby falsify its description! This last achievement deserves, in my proposal, the name of ‘Airy’s paradox’.”

Starting Funds


Three men play a game, agreeing that in each round the loser will double the money of each of the other two. After three rounds, each man has lost one time, and each man has $24. How much did each have at the start of the game?

Click for Answer

A Geometric Illusion

geometric illusion

Which of the two shaded areas is larger, the central disc or the outer ring?

Surprisingly, they’re equal. Each of the concentric circles has a radius 1 unit larger than the last. So the area of the central disc is π × 32 square units, and the area of the outer ring is π × 52 – π × 42 = π × 32 square units. So the two areas are the same.