One other interesting item from Paul Halmos’ Problems for Mathematicians, Young and Old (1991): Pick a point in the first quadrant and draw a downward-sloping line through it. This line makes a triangle with the coordinate axes. At what angle should we set the line to minimize the area of the triangle?

This problem yields to calculus, but there’s a simple geometric solution. Reflect the axes through the point to make a box:

halmos minimum

Now as we swivel our line through the point, it defines two triangles, one against each set of axes. The area of the combined triangles is equal to or greater than the area of the box. So, intuitively, it reaches a minimum just as the swiveling line becomes a diagonal of the box. That’s the answer.


Harry Mathews assembled lines from 14 existing sonnets to make a new one:

Shall I compare thee to a summer’s day
And dig deep trenches in thy beauty’s field?
Why lov’st thou that which thou receiv’st not gladly,
Bare ruin’d choirs where late the sweet birds sang?
Anon permit the basest clouds to ride
And do whate’er thou wilt, swift-footed Time:
Nor Mars his sword, nor war’s quick fire, shall burn
Even such a beauty as you master now.
Love’s not Time’s fool, though rosy lips and cheeks
(When other petty griefs have done their spite,
And heavily) from woe to woe tell o’er
That Time will come and take my love away;
For thy sweet love remembered such wealth brings
As any she belied beyond compare.

“This new poem sheds light on the structure and movement of the Shakespearean sonnet,” he wrote. “Nothing any longer can be taken for granted; every word has become a banana peel.”

(Harry Mathews, “Mathews’s Algorithm,” in Warren F. Motte, ed., Oulipo: A Primer of Potential Literature, 1998.)

Good for the Gander

In 1923, Ollie Kraehe, owner of the NFL’s St. Louis All-Stars, approached Green Bay Packers coach Curly Lambeau with a tempting offer: He would give him end Jack “Dolly” Gray in exchange for some cash that he needed to fund his team. Lambeau leapt at the deal: Gray was reputed to have been an All-American honoree at Princeton in 1922 and sounded like the best player on Kraehe’s team.

Two weeks later, Lambeau cornered Kraehe and demanded an explanation — in his first game with the Packers, Gray had played terribly.

Kraehe told him the truth: The man had approached him earlier that year, representing himself as a Princeton star, but he played so badly that after three games Kraehe had investigated and found no such background — he was simply an impostor. Kraehe had traded him on to Lambeau as a joke, letting him coast on his phony reputation and expecting to take him back when the joke was over. But by that time the impostor had disappeared. His identity remains unknown.

Image: Wikimedia Commons

Hidden on the back of the National World War II Memorial in Washington, D.C., is an engraving of Kilroy, the ubiquitous graffito that accompanied American GIs through Europe and, later, around the world. Some earlier inscriptions:

Istanbul’s Hagia Sophia has stood since 537, built by Justinian I as the patriarchal cathedral of Constantinople. It wasn’t until 1964 that runic inscriptions were discovered in the southern gallery, apparently engraved by members of the Varangian Guard during the Viking Age. Their meaning isn’t certain, but one may have read “Halfdan carved these runes” and the other “Ári made the runes.” More may yet be found.

The Piraeus Lion, one of four marble lion statues now at the Venetian Arsenal, bears runic inscriptions apparently made by Scandinavians in the 11th century, when it had stood in Athens. One reads, “Asmund cut these runes with Asgeir and Thorleif, Thord and Ivar, at the request of Harold the Tall, though the Greeks considered about and forbade it.” The other reads, “Hakon with Ulf and Asmund and Örn conquered this port. These men and Harold Hafi imposed a heavy fine on account of the revolt of the Greek people. Dalk is detained captive in far lands. Egil is gone on an expedition with Ragnar into Romania and Armenia.”
Images: Wikimedia Commons

Desire for a Monument

Set a monument for me,
built of sugar, in the sea.

It will melt, of course, and make
briefly a sweet-water lake;

meanwhile, fishes by the score
take surprised a sip or more.

They, in various ports, will then
be, in turn, consumed by men.

This way I will join the chain
of humanity again,

while, were I of stone or steel,
just some pigeon ungenteel,

or perhaps a Ph.D.
would discharge his wit on me.

— Christian Morgenstern

Snow Manipulation

A puzzle by James M., an operations researcher at the National Security Agency:

Frosty the Snowman wants to create a small snowman friend for himself. The new snowman needs a base, torso, and a head, all three of which should be spheres. The torso should be no larger than the base and the head should be no larger than the torso.

For building material, Frosty has a spherical snowball with a 6-inch radius. Since Frosty likes to keep things simple, he also wants the radius of each of the three pieces to be a positive integer. Can Frosty accomplish this?

Click for Answer

In a Word

v. to watch carefully; to keep vigil

n. lateness; delay

adj. roving; wandering; characterized by vagaries

adj. that roams around the world

Letter to the Times, Aug. 24, 2001:

Sir, Some years ago when I lived in Livingstone, Zambia, I received a letter from Lusaka, the capital, correctly addressed but weeks late.

Among the numerous back stamps the envelope had collected in its travels the last was Livingstone, Canada, with an inscription ‘Try Livingstone, Zambia’.

The letter had also visited Scotland and New Zealand.

Yours truly,

David H. Walton
Crowland, Peterborough

Image: Wikimedia Commons

Drive east on Canusa Street and you’re in the United States, drive west and you’re in Canada. That’s because the street, part of Quebec Route 247, runs along the border — the yellow line dividing its lanes follows the 45th parallel.

Families who live on the south side are in Vermont, and those in the north are in Quebec — they can see one another but must present themselves at a border post before crossing the street.

But residents of both countries can freely visit the Haskell Free Library — where the main entrance is in America but the books are in Canada.

Man of the World

Names of the Mad Hatter in various translations of Alice’s Adventures in Wonderland:

  • the Hatmaker
  • the Maker of Hats
  • the Hatman
  • the Man Who Made Head Protection
  • Mr. Tophat
  • Owl
  • Master Hats
  • Marble Mason
  • Stockman
  • Blockhead
  • Baboon
  • Fellow With Hats
  • Cap-Wearing Person
  • Kynedyr Wyllt mab Hettwn Tal Aryant

That last one’s in Middle Welsh. Though Lewis Carroll’s novel abounds in wordplay, rhymes, quotations, nonsense, homophones, logical twists, and Victorian allusions, it’s found its way into 174 languages and more than 9,000 editions around the world. Zongxin Feng of Tsinghua University in Beijing wrote, “Of all Western literary masterpieces introduced into China in the twentieth century, no other work has enjoyed such popularity.”

In an 1866 letter, Carroll had written, “Friends here [in Oxford] seem to think that the book is untranslatable.”

(Jon A. Lindseth, ed., Alice in a World of Wonderlands, 2015.)


  • The Dutch word for cease-fire negotiations is wapenstilstandsonderhandelingen.
  • Rearrange the letters in ONE THOUSAND KILOS and you get OH, SOUNDS LIKE A TON! (Hans-Peter Reich)
  • 1167882 + 3211682 = 116788321168
  • The Irish for chess, ficheall, derives from the Old Irish fidchell, “wood intelligence.”
  • “Life is a school of probability.” — Walter Bagehot

A tiny detail that I hope is true: In Time in World History (2019), historian Peter Stearns writes that before watches became affordable, some European soldiers “took their own roosters with them so they would wake up on time.”