This morning I’m about to get on the T and go to Cambridge City Hall for a significant personal event. I’d like to celebrate the occasion with a few lines of verse by the great mathematician (and lesser poet) James Joseph Sylvester. (A tip of the top hat to Jerry Alexanderson for bringing Sylvester’s versifying to my attention.)

Fairest, O ! of lily-kind,
Perfect pearl and priceless find !
Pure as poet’s milk-white hind,
Spirit ! from all dross refined,
Hearts to ravish Heav’n-designed ;
Fresh as rills that sparkling wind
On their way the sea to find,
[O'er smooth pebble-bed patíned.]
Dew-drop ! on which sun hath shined,
Shedding glamour undefined,
And a light that doth remind
of the cloudless heav’n behind.
Whose light laugh, like bells sweet-chimed,
[Or hushed trill of bow refined
Touched with fingers taper-tined,]
Piercing feeling’s inmost rind,
Tone-spray tossing on the wind
Leaves no smart unmedicined.

  With fond folly all untined,
  To each duteous law resigned ;
    (Round her finger she can wind
    All to obey her disinclined :
    Who so wilful or so blind
    To say nay to Rosalind !)

Riffling through some file folders last night, I happened upon an item that I evidently clipped out of Datamation years ago. It’s titled “Magic Moments in Software,” by Deborah Sojka and Philip H. Dorn. I can’t find a date on any of the pages, but internal evidence suggests it’s from the early 1980s.

The piece is organized as a timeline, listing various notable events in the history of software, such as the first successful run of a “user-written, meaningful Fortran program” (20 April 1957), the release of Lisp (1962) and BASIC (1964), and the publication of The Mythical Man-Month (1975). One of the early entries caught my eye:

1949—F. Fortesque Fingerhut, while trying to debug his first program on the ACE computer at the National Physical Laboratory, cannot find the problem. He cracks under the strain, disappears, and is not seen until 1981 when he reemerges as the net court judge at Wimbledon.

All the other events recorded in this chronology seem to be genuine, and there’s no April Fool disclaimer at the end. Is it possible that the story of F. Fortesque Fingerhut is not a joke?

Catching up on new submissions to the arXiv, I came across a paper by Robert Baillie, “Summing the Curious Series of Kempner and Irwin,” which is item 0806.4410 in the mathematics listings. Here’s the abstract, exactly as it appears at http://lanl.arxiv.org/abs/0806.4410v1:

In 1914, Kempner proved that the series 1/1 + 1/2 + … + 1/8 + 1/10 + 1/11 +… + 1/18 + 1/20 + 1/21 + …, where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. (The actual sum is about 22.92068.) In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9’s is also a convergent series. We show how to compute sums of Irwins’ series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39+ 1/49 + … where the denominators have exactly one 9, is about 23.04428708074784831968. Note that this is larger than the sum of Kempner’s “no 9″ series. We also show how to construct nontrivial subseries of the harmonic series that have arbitrarily large, but computable, sums. For example, the sum of 1/n where n has at most 434 occurrences of the digit 0 is about 10016.32364577640186109739.

Baillie’s article is full of really interesting mathematics and algorithmics, which ought to be reason enough to mention the paper here. But it was something stranger that caught my attention. Look closely at the large number in the last line of the abstract. The HTML source of the arXiv page looks like this:

  1<a href="abs/0016.3236">0016.3236</a>4577640186109739.

It seems the sequence 0016.3236, embedded in a larger string of digits, has been interpreted as an arXiv identifier. I can only guess that some program at arXiv.org is scanning abstracts looking for strings of the form nnnn.nnnn, where n is any decimal digit. It’s a little like those rogue search-and-replace scripts that do amusing things like turning every “gay” into a “homosexual.”

As it happens, there is no arXiv paper with the identifier 0016.3236. There can’t be because the identifier format is actually yymm.nnnn, encoding the year and month of submission in the first four digits. Obviously there is no month 16, and the identifier scheme had not yet been introduced in the year 00. Thus, as far as I know, the competition is still open for the first perfectly self-referential arXiv preprint—one that finds a legitimate reason to embed its own identifier in its abstract. (Part of the challenge is that you can’t know in advance—at least not with high precision—what number will be assigned to a paper when it is submitted.)

[More on the Kempner series: Baillie and Thomas Schmelzer also have an article on related work in the latest American Mathematical Monthly. The article is available (pdf) from Schmelzer's web site.]

Update 2008-09-07: Because this post makes fun of programs that automatically scan and “fix” text, it inevitably suffers that very fate. Although the long quotation above was pasted into my editor window exactly as it appeared in the original arXiv abstract, that’s not how the quotation will arrive on your screen. If you try to follow the link embedded in that long number, you’ll find that the address is not “abs/0016.3236″ but “bit-player.org/2008/abs/0016.3236″. Somewhere in the WordPress software is a routine that “corrects” any URL lacking a top-level domain. I have resisted the urge to correct the correction. I note that arXiv is following the same policy: The latest version of the Baillie abstract (posted 17 August) still includes the curious embedded link.

Update 2009-07-31: In the 2008-09-07 update I unjustly impugned WordPress. It is not WordPress that is munging the URL; it is my very own browser (and probably yours too). A URL without a fully qualified domain name is interpreted by the browser as a link relative to the current site. Viewing the page source shows that the “abs/0016.3236″ is not changed within the HTML. Many thanks to Keith Beckman for setting me straight on this.

I’ve been Scrabbling by email lately. In today’s game my partner started out by playing

                    H
                    E
                    X

and I responded with

                   A H
                   W E
                   E X

At this point my opponent might well have continued with another three-letter word to make a tidy square block such as:

                  Y A H
                  E W E
                  S E X

In actuality she did something quite different. She played a seven-letter “bingo,” using all her letters to earn a 50-point bonus; as a result I’m hopelessly far behind in the scoring. But let us say no more about the tawdry details of winning and losing; there’s a puzzle here. Looking at that three-by-three block of letters and words, it occurs to me there must surely be legally reachable configurations of a Scrabble board that have no legal continuation. Scrabble rules say that, except for the first move, letters can be added to the board only on squares adjacent to existing letters, and all sequences of two or more letters (both vertically and horizontally) must be dictionary words. The rules say nothing about the situation where continued play is impossible.

I’m sure there must be many stymied positions, where no words can be formed, regardless of what letters you have on your rack. Or so I assert; but, the fact is, I haven’t been able to find even one such configuration. A cursory examination of the list of all allowed two-letter words argues that no two-by-two block of letters can be stymied. What about two-by-three or three-by-three blocks? Somebody must have settled these questions, but my Googling has failed to find the answer. What is the smallest stymied position? (I don’t require that a solution be a rectangular block of letters, but having stray letters dangling off the edges of a block makes it easier rather than harder to form words.)