Last Wishes

British Army gunner Frank Bracey wrote this letter to his wife in May 1916 and left it to be opened in the event of his death:

Dearest Win

I am writing just a line Win in case of accidents. Just to let you know how I have always loved you Dear. You are the best little girl on God’s earth have I told you before. But I am writing this because I have a feeling that I shall not come back again. I have most of your letters in this box Dear and I wish you to have them and the cards. You may think I am a bit taped writing this dear but I cannot help it. If I do come back dearest you will never see this letter but I have a strong feeling that I shall never see England again. In case I do pop under the earth I want you to be happy and look out for a worthier chap than your Humble, you have been every thing to me Win. I know your love is mine forever dearest but if I do not come back I wish you the best of happiness and a good husband. I know you told me what you would do for yourself if I did not return but Win for the sake of our love I wish you to be brave, it would be hard for you little girl I know, but do nothing of the kind. My last wish is that you marry a good man and to be happy and to think of your Humble now and then. I felt I must write these few lines Win but whatever happens dear just keep a stout heart and think that your Frank did his bit for the women of this little isle. I expect you will think your Humble crazy but I was never saner than I am now.


He was killed in Pas-de-Calais that August. He is buried at the British military cemetery at Saint-Amand.


When David Huffman died in 1999, the world lost a talented computer scientist — Huffman was best known for discovering the Huffman coding technique used in data compression.

But it also lost a pioneer in mathematical origami, an extension of the traditional art of paper folding that applies computational geometry, number theory, coding theory, and linear algebra. The field today is finding wide application, helping researchers to fold everything from proteins to automobile airbags and space-based telescopes.

Huffman was drawn to the work through his investigations into the mathematical properties of “zero curvature” surfaces, studying how paper behaves near creases and apices of cones. During the last two decades of his life he created hundreds of beautiful, perplexing paper models in which the creases were curved rather than straight.

But he kept his folding research largely to himself. He published only one paper on the subject (PDF), and much of what he discovered was lost at his death. “He anticipated a great deal of what other people have since rediscovered or are only now discovering,” laser physicist Robert Lang told the New York Times in 2004. “At least half of what he did is unlike anything I’ve seen.” MIT computer scientist Erik Demaine is working now with Huffman’s family to recover and document his discoveries (PDF).

“I don’t claim to be an artist. I’m not even sure how to define art,” Huffman told an audience in 1979. “But I find it natural that the elegant mathematical theorems associated with paper surfaces should lead to visual elegance as well.”

One-Sided Story

In the runup to Thailand’s 2001 elections, Thai Rak Thai party founder Thaksin Shinawatra faced allegations of corruption. The Bangkok Post‘s “week in review” email examined the charges against him, his attempts avoid the media, his reputation, and the Internet’s reaction. It used these paragraph headings:

Thaksin cited
Thaksin sighted
Thaksin slighted
Thaksin sited

In Indexers and Indexes in Fact and Fiction, Hazel K. Bell writes, “Clearly the editor is an indexer manqué.”


langstaff chess puzzle

W. Langstaff offered this conundrum in Chess Amateur in 1922. White is to mate in two moves. He tries playing 1. Ke6, intending 2. Rd8#, but Black castles and no mate is possible. But by castling Black shows that his last move must have been g7-g5. Knowing this, White chooses 1. hxg6 e.p. rather than 1. Ke6. Now if Black castles he can play 2. h7#.

“Not so fast!” Black protests. “My last move was Rh7-h8, not g7-g5, so you can’t capture en passant.”

“Very well,” says White. “If you can’t castle, then I play 1. Ke6.” And we’re back where we started.

“What was really Black’s last move?” asks Burt Hochberg in Chess Braintwisters (1999). “If a position has a history, it can have only a single history, and Black would not be able to choose what his last move was any more than I can choose today what I had for dinner last night.”

“This is not a real game, however, but a problem in chess logic. The position’s history does not exist in actuality but only as a logical construct.”

In a Word

adj. long-winded

[Edmund] Falconer’s Oonagh, or, The Lovers of Lismona opened one evening in 1866 at half-past seven. By midnight most of the audience had left; by two o’clock in the morning only a few sleeping critics were still there. At three o’clock the stage crew brought the curtain down with the action still in progress and the play was taken off.

The Methuen Drama Dictionary of the Theatre, 2013

Beatty Sequences

Here’s another interesting source of complementary sequences. Take any positive irrational number, say \sqrt{2}, and call it X. Call its reciprocal Y; in this case Y = 1/\sqrt{2} = \sqrt{2}/2, or about 0.7. Add 1 to each of X and Y and we get

1 + X ≈ 2.4

1 + Y ≈ 1.7.

Now make a table of the approximate multiples of 1 + X and 1 + Y:

beatty sequences

If we drop the fractional part of each number in the table, we’re left with two complementary sequences — every number 1, 2, 3, … appears in one sequence or the other, but never in both.

They’re called Beatty sequences, after Sam Beatty of the University of Toronto, who discovered them in 1926. A pretty proof by A. Ostrowski and J. Hyslop appears in the March 1927 issue of the American Mathematical Monthly and in Ross Honsberger’s Ingenuity in Mathematics (1970).

A House Afire

If box-office receipts are adjusted for inflation, Gone With the Wind is still the highest-grossing film of all time, with earnings of $3.4 billion in 2014 dollars.

After the film’s 1939 premiere in Atlanta, playwright Moss Hart wired producer David O. Selznick:


A Hidden Economy

During the American Civil War, enemy soldiers would sometimes meet to barter. Tobacco was hard to get in the North, and coffee was scarce in the South, so, where it could be done safely, soldiers would meet between the lines to trade.

In some cases this was done across distances. If a river or lake separated the lines, a tiny boat would be laden with commodities and sent to the other side, where it would be unloaded and filled with exchange cargoes, as agreed on by shouting and signaling across the water. On the Rappahannock early in 1863 a group of New Jersey soldiers received a shipment “by miniature boat six inches long.” It carried this note:

Gents U.S. Army

We send you some tobacco by our Packet. Send us some coffee in return. Also a deck of cards if you have them, and we will send you more tobacco. Send us any late papers if you have them.

Jas. O. Parker
Co. H. 17th Regt. Miss. Vols.

Alfred S. Roe, who served in a New York artillery unit, recalled that near Petersburg in the winter of 1864, “a certain canine of strictly impartial sentiments” was “taught to respond to a whistle from either side. Thus with a can of coffee suspended from his neck he would amble over to the Johnnies, and when they had replaced coffee with tobacco he would return in obedience to Union signals, intent only on the food reward both sides gave him.”

(From Bell I. Wiley, The Life of Billy Yank: The Common Soldier of the Union, 1952.)