Draw a 4-by-4 grid. Colour each square red or blue. Select any square, S, and write 1 in it. Then write 1 in every square you can reach from S by a series of moves, where each move is from a square to an adjacent, horizontally, vertically, or diagonally, square of the same colour.

Select any empty square, T, and write 2 in it. Then write 2 in every square you can reach from T by a series of moves. Repeat this procedure for 3 and then 4 and so on until every square has a number. The last number you write down is called the “score” for that colouring. If we imagine the grid is the map of a farm, then you have divided the map into fields, one field for each number.

(1) What is the largest score possible?

(2) If we work with a 5-by-5 grid, what is the largest score possible?

In brief: What is the maximum number of queenwise-connected monochromatic regions you can make by coloring each square of an \(n\times n\) grid either red or blue? If we were asking about rookwise-connected regions, the answer would obviously be \(n^2\). But queenwise-connected is at least slightly less obvious!

For the 5-by-5 grid, the “obvious” answer is

rBrBr
BBBBB
rBrBr
BBBBB
rBrBr

for a total of 10 regions. But the answer for 4×4 is not simply

rBrB
BBBB
rBrB
BBBB

for a total of 5; you can in fact make 6 regions! (Once you’ve seen the trick it’s hard to unsee; but I didn’t see it at first myself.)

Discussion on the SeqFan mailing list tentatively produced the sequence: 1, 2, 5, 6, 10, 12, 17, 19, 26, (28?)… Here’s the pattern for odd \(n\), and a conjectured pattern for all even \(n>4\).

For odd \(n\): \(\lceil n/2\rceil^2 + 1\) regions. For even \(n>4\): \(\lceil n/2\rceil^2 + 3\) regions.

was ignorant of the dispute two years ago when he developed a competitive crossword game called Pago Pago and first became aware of crossword fans’ passion for the pastime.

“Guilbert invented the crossword-based game Pago Pago” is repeated in Thomas Frisbie, “Fan passes word on puzzle hall of fame” (Chicago Sun-Times, 1988-08-14). (Thanks to Ben Zimmer for identifying the article!) But that’s all I can find in print about Pago Pago.

Helene Hovanec’s article “Robert Guilbert’s Crossword Academy” (CROSSW RD, Nov/Dec 1992) contains no text about Pago Pago, but does include this uncaptioned photograph of Guilbert (1911–1990) hard at work:

David Steinberg of the “Pre-Shortzian Puzzle Project” blog has mentioned Guilbert several times in connection with Guilbert’s short-lived Crossword Academy, and even received in 2016 a very nice blog comment from Guilbert’s son Jonathan and a little more via email. Jonathan wrote, in part:

In the photo he can be seen working on a puzzle that he designed as a “Moebius Strip” — an “Infinity Crossword Puzzle” — a puzzle without a beginning and an end. He was going to have this game manufactured and marketed under the name “Pago Pago,” I believe.

Personally I suspect that Jonathan’s hunch was wrong: that the Möbius-strip puzzle and the “Pago Pago” puzzle-game (whatever its nature was) were two separate projects. However, apparently on the strength of Jonathan’s guess, the Möbius–Pago Pago hypothesis was reproduced as fact in Adrienne Raphel’s dissertation “The Crossword Mentality in Modern Literature and Culture” (Harvard, 2018): “In 1986, Robert Guilbert, a former ad copywriter from Milwaukee who enjoyed spending his retirement doing the puzzle, invented a game called Pago Pago, an infinite crossword on a Möbius strip, and began to peddle it around to crossword fans.” The same sentence appears in Raphel’s book Thinking Inside the Box (2020). The game of telephone continues in Natan Last’s Across the Universe (2025): “he’d invented a game called Pago Pago, an infinite crossword on a Möbius strip, and wanted to hawk it to the greats.”

UPDATE, 2026-08-13: Actually, Guilbert had already gotten Pago Pago into circulation long before these articles were written! The July 1986 issue of GAMES magazine has a review of Pago Pago (Garden City, New York: Just Games). Indeed there’s no Möbius element; the game is simply a pack of 24 William Lutwiniak crosswords plus rules for up to four players to complete a set of grids competitively: fill squares for three minutes, then pass the grid to your left and fill squares in your opponent’s old grid. Whoever fills in the most squares correctly (“with penalties for errors and bonuses for correcting opponents’ errors”) is the winner. The game has an entry on BoardGameGeek, and as of this writing you can buy a copy for $25 on eBay. (Thanks to Virginia Downes for this information!)

But what does a crossword on a Möbius strip look like, anyway? said I to myself.

Presumably it would be a crossword that loops around on itself from bottom to top (and vice versa). A crossword that does that on all four edges would be “toroidal,” like this one (which I found on Keith Calkins’ “tori crosswords” page, although I infer that Jeff Weeks was the author):

Jeff Weeks has made a bunch of these; see this PDF from MoMath. In fact he’s even made crosswords “on a Klein bottle.” Here the right and left edges connect as before, while the top and bottom edges connect with a Möbius half-twist.

The above crossword is neat because despite being 5x5, its two longest entries are 9 letters each. But it’s less than perfectly satisfying in that half of the Across entries read right-to-left; and in that if it actually wrapped around on itself with a half-twist, wouldn’t the letters wind up backwards too? Then 6-Down wouldn’t be MESSENGER but rather MESSENGER, and so on. (2-Down mirror-reverses very nicely, though!)

Now, a real mathematical Möbius strip, says Martin Gardner,

is what topologists call “nonorientable.” Imagine the strip to be a true surface of zero thickness. Embedded in this 2-space are flat creatures that are mirror-asymmetric (not identical with their mirror images). If such a creature moves once around the band to rejoin his fellows, he will have changed his parity and will have become a mirror reflection of his former self.

If our “flat creatures” are the letter-forms of the crossword entries, then when the crossword loops around on itself the letter-forms will change parity: S becomes S, and so on. So the only letters we seem to be able to use are the vertically symmetric ones, and we’ll have great difficulty constructing any crossword more complex than the following infinite jest:

Or we could loop horizontally with horizontally symmetric letters (HEEHEEHEE…). But to create entries that aren’t infinitely long, we should create a diagonal “ladder” of words instead — as I fancy I see Robert Guilbert doing in that photo! If our “ladder” runs diagonally down and left, like this:

  ABC
 DEF
GHI

then we’ll be able to use letters whose axis of symmetry is on that 45-degree line, such as (of course) O and X, but also L: “LOX” horizontally will flip to produce “XOL” as one of the Down entries. In fact we can also use W, which (if you draw it just right) flips to become E; H, which flips to become I (with serifs); C, which flips to become U; and even D, which (with long serifs) flips to become a nice round capital A. This gives us enough degrees of freedom to create the following Möbius-strip crossword:

Print out this strip and tape the ends together with a half-twist. Fill it out with a heavy ink marker and you’ll find you only need to solve the first five clues — then you can flip the strip over and read off the answers to the last five clues in the letter-forms that have bled through the paper! (Or hold the back of the paper up to the light; or, for the really lazy, just lay a mirror alongside it.)

Mine is not a good crossword. Not only does it have awful fill, it repeats entries: six of the Acrosses are repeated among the Downs. But it was the best three-letter ladder I could find even with computer assistance. My dictionary of possible answers consisted of only 23 pairs of words. (Remember, every Across entry flips into a Down entry: if we have LAW across, we necessarily get EDL down. And since EDL isn’t a word, LAW isn’t in our dictionary either.) Words mirrorable this way of more than three letters are rare as hen’s teeth: HOWL flips into LEO I, ALOE into WOLD, and ALOUD into A COLD; that’s about all.

Can you come up with a better Möbius-strip crossword? or do you have additional information on Robert Guilbert’s Pago Pago and Möbius-strip projects? Email and tell me about it!

See also:

• “Fully Schrödinger crosswords” (2025-01-03)

Ford’s canto, like Dante’s, is exactly 136 lines of pure terza rima, containing a few impressively novel rhymes and many powerful images. Notably Ford finds room to expand Dante’s Oh felice colui cu’ ivi elegge! from one line into three.

Just halfway through my own trip in our living,
Provisional, short world, I found that I
Had lost my path — woods dark and unforgiving
Making my way so hard that I must sigh
If I would say how brutal was that wood,
To think on which will always horrify.
If anything, it’s only dying could
Outdo that acrid wood’s malignity.
But I will turn from horror, towards that good
Which also on that pathway, luckily,
I found, though I can’t say what hid all truth
Away in total blank stupidity.
I cannot work out how it was, in sooth,
That in that gloom awaking, full of stupor,
I’d lost that saving road I took in youth.
I got at last to what you’d call a croup or
Boundary of that dark and dismal hollow,
A bank of rocks transforming hill-wall to floor.
To that bluff’s scapula my look did follow
Upwards still that guiding morning glow,
Bright from that star so holy to Apollo.
Thus did that anxious horror, which had so
Brought churning with it, all that awful night,
To my soul’s pool, diminish, calm its flow.
And as a man who pants hard, still in sight
Of billows which, almost, brought him to sink,
From land looks back on that main, full of fright,
So did my spirit, panicking, still think
On that grim pass no mortal human can
Go out of without passing living’s brink.
At last my body pausing for a span
Until invigoration from that stop
Could now allow it motion, though no plan
Could show it how to go, with constant prop
On foot in back and downwards, sought to gain,
By climbing always upwards, that mount’s top.
And lo! A cougar stood stock still, though plain
Its quick and light agility. No turning
Could pass it by. Upon its skin a rain
Of spots, which I, that bright and airy morning
(As sun and star-companions Loving God
Had first spun, still did spin) saw as adorning
That cat, a sign of succor I might laud,
During that blissful dawn, but not so bright
That I did not start shaking, on that road,
At what was now arising in my sight:
A lion, drawn up high, intimidating,
So much that air and I, both full of fright
About how it its stomach might start sating,
Must stop — for what? A wolf; and I was pavid,
So skinny was it, though anticipating
It would sup soon on anything, for avid
It was for food: though lank it was full too:
That wolf so ruinous to man was gravid!
And I was too — with fright! I could not do
What I was hoping for — to climb that hill.
Alas, I found no pathway round nor through.
And as a man who’d got just what his will
Saw as most worth wishing of all things,
Now gasps on losing it, a poignant thrill
Of brutal pain, transfixing with its stings
From that wolf, did I sob at, and I shrank
Darkwards, away from Sol’s loud blazonings.
Back did I go, back to that lowland dank.
Abruptly in my sight shows up a man,
Who, dumb so long, I thought would sound as blank
As our surround was. Straight to him I ran,
Still crying “Pity!” to him, “Man or shadow
Of a man!” “Not a man, but Mantuan,
By birth,” was his account, “born sub Julio,
But it was good Augustus, though his gods
Did not say truth nor know your Christian Trio,
Who was my king. My song got many nods,
All praising it as Roman history,
From Trojan loss to gain, against all odds,
From burning Ilion to victory.
But why do you avoid that joyous mountain,
As though to climb you’d no ability?”
“Now art thou Virgil? That riparian fountain,”
All blushingly I said, within his sight,
“Of words that all who follow find a sound in
Which our own songs would sing too? If I might,
I’ll say how much I honor you, what study
I sanctify with loving to your light.
But now look on that animal so bloody,
To you I turn for aid, to you I’m flying.
O horror!” “You must go this path, though muddy,”
Did Virgil say, to try to hush my crying,
“If you would from this dark wood find a way.
For past that wolf can no amount of trying
Attain that goal; whoso will try will pay
With his annihilation; hungrily,
Voraciously that wolf puts him away.
Marrying many animals bodily,
It looks to go on doing so, until
A Sighthound, coming with finality,
Shall bring its pain to culmination, kill
That awful wolf, who did not pity show.
That hound, consuming not, such is its will,
Things of this world, but only what would go
With wisdom — loving all morality —
In fabrics of Franciscan monks will know
His nation. Savior of low Italy
That hound is, land for which Camilla, dying
A maid, and Nisus, and his loving ally,
And Turnus, all lay down for good. Now plying
Its way through any town that it might harry,
That wolf cannot avoid damnation, buying,
Through vicious rivalry, a day to tarry,
So soon to finish. So I think it right
That I conduct you, and that you stay wary
As onwards through this always-lasting night
You go, with sounds of always-lasting sorrow
Imploring total dying, not this blight.
But going through, I will bring you tomorrow
To souls who, though in pain, stay happy with it,
Hoping to pay back soon what sin did borrow
And climb salvation’s mount to that first orbit
To which I must not go. But if you will
You’ll find at that hill’s top a worthy spirit,
Which I am not, as I did not fulfill
Writs laid down by that just, all-causing King,
Combatting what was law. His codicil
About my task thus says: I may not bring
A pilgrim to his city. Though his might’s
Ubiquitous, from that city starts its ring
Circling all worlds. O happy, any sights
Of Him; most so, who in that city strong
Find bright salvation in its million lights.”
And I to him: “I pray, by your high song —
And by that God you did not know, I pray —
That you will as conductor, for as long
As you can do so, bring us on that way
Of pain, at last to portals which saints hallow,
And far from this soil’s sorrow.” With no stay
On did Virgil go; downwards did I follow.

  —Kinton Ford, 2020 or 2021

Editorial notes: I adjusted some little punctuation relative to the Exquisite Pandemic text. For example I removed a comma before “About how” and a period after “stay wary,” changed a semicolon after “dank” to a period, and inserted commas around “though in pain.”

Though I didn’t alter them above, the confusing lines “a poignant thrill / Of brutal pain, transfixing with its stings / From that wolf, did I sob at, and I shrank / Darkwards” seem to detach the phrase “from that wolf” a bit too much from “I shrank.” One might take this as a cleverly jarring bit of textual iconicity (“I” literally retreating behind the “sob” to get away from the “wolf”), and/or one might prefer to imagine something more straightforward like “…stings, / Brought sobs; and, wolf advancing, so I shrank” or “…stings, / Shot through my gut, and from that wolf I shrank.” I also find some disorder in the sentence beginning “Upon its skin a rain,” but I’m less sure what might be done about that or even whether anything ought to be done, beyond maybe changing the comma after “dawn” to a semicolon.

The amount and quality of intelligence utilized in the working out of cross-word puzzles almost precisely equal those that were revealed some years ago when an amazing — and discouraging — army of Americans filled the newspapers that cater to such folk with triumphant solutions of an arithmetical problem concerning the age of a supposititious Ann — a problem that any eighth-grade boy or girl who is “all there” should have solved in his or her head in about thirty seconds.

The older I get, the more sense it makes when the phrase “some years ago” turns out to refer to something from two decades back. The problem, apparently first published in the New York World circa 1903-10-09,^[1] goes like this:

Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?

(A writer to the Montgomery (Alabama) Advertiser of 1903-10-24 points out that you can get the apparently-most-common wrong answer if you read the puzzle incorrectly as “She was twice as old as Ann was when Mary [sic] was as old as Ann now is.”)

The New York Evening Post (quoted in the Princeton (Minn.) Union) opined:

“Wherever it may have originated — and this is itself a matter of controversy — the problem concerning the ages of Ann and Mary has caught the country’s fancy as few such things have since ‘Pigs in Clover.’ ”

Billboard (1904-02-20): “The question ‘How Old Is Ann?’ has been dramatized and was staged at White Plains. Barney Gerard, the author, makes Ann a mischievous little girl, who refuses to go to school. The school authorities get after her to punish her but must first find out how old she is. Complications result.” • The New York Evening World (1903-11-17): “ ‘How Old Is Ann?’ is the title of a new song which Frank Daniels sang last night at the Victoria Theatre. When the opening lines sounded the audience settled back in its seats with a sigh of relief. ‘At last,’ it meant just as if it was expressed in words, ‘we are going to find out how old that cussed sister of Mary is.’ But they didn’t, and there’s where the fun came in. The song has a rattling chorus and a catchy tune.” • Cartoonist Ed F. Flinn fingered Ann as the cause of the Panamanian secession. • The World of 1903-11-07 “revealed” the perpetrator of the problem as one “Mr. Perkin Warbeck, of Newark, N.J.”

The Philadelphia North American published “the answers of a few leading Philadelphians.”

“I don’t see why that simple question is puzzling anybody. I knew Ann is 12 years old as soon as I saw the problem,” said Frank R. Shattuck. “There is no other solution, I tell you. Look here, I’ll explain it to you. Mary is 24 years old. She is twice as old as Ann — that’s a cinch. Of course, then, Ann is half Mary’s age, or 12 years. There ought to be a comma after ‘Ann,’ though.”

It was suggested to Mr. Shattuck that there was no comma in the problem.

“Doesn’t make any difference,” replied he promptly. “Let’s see — twice as old as Ann ‘was.’ That ought to be ‘is’ — as Mary was when Ann is. I’m not mixed up. I tell you, Ann is 12.”

Not to be outdone, the New York Evening World published on 1903-11-09 and 1903-11-10 the answers of “many prominent men of New York,” including:

Policeman Bert Gallagher: “Ann is old enough to know better than not tell her age when she knows how much trouble her silence is causing. She ought to be ashamed.”

John T. Meehan: “Ann was ninety years old. I know, because she and her brother shook hands with Lafayette on his visit to New York in 1824. Old Peter was the steward of the beanery for forty years, and he was ninety-two when he died; Ann was three years younger. He and Ann used to sell newspapers on the Vesey Street corner during Jackson’s first term. Why, Ann was— Oh, that World question about the girl who’s twice as old as herself? I haven’t time to fuss with that. I thought you were talking about old Peter Rose’s sister Ann.”

Timothy P. Sullivan: “Just twice as old as she was when Mary was twice as old as Ann. That’s easy. Six years ago Mary was eighteen and Ann was thirteen. To-day Mary is twenty-four and Ann is nineteen. Figure it out for yourself. Don’t you see that if Mary is twice as old as Ann was when Mary was— I mean twice as old as Mary when Ann was half as old as Mary, consequently— but there, let the professors figure it out. It’s as bad as ‘when is a mice when it spins.’ ”

The Wyoming Tribune of 1903-10-28 calls it “No Puzzle At All”:

Anyone could figure it out if they would stop to think. Ann is 16, of course, because if Mary is twice as old as Ann was, all you have to do is to find out when. Now she is 24, Ann’s age was 12, or twice six. Mary is now four times six, which being divided by six, gives four. Add this to twelve, and there you are. —Smith.

adding:

When Mary had a little lamb, she was twice as old as Ann was when Ann was as old as Mary now claims to be. How old is the sheep? —James Barton Adams.

A correspondent for the Kansas Journal wrote:

Constant Reader,
Ann is sixty-three years of age. Obviously, Ann is as old now as Mary was when Mary was half as old as her present age less the age of Ann at the time she was half as old as Mary was when she was twice as old as Ann is, unless you figure that Ann was half as old as if she were twice Mary’s present age, less Ann’s age when she was twice as old as Mary is now, or twenty-four. Twenty-four and fifteen are thirty-nine, and twenty-four and thirty-nine are sixty-three, which represents Mary’s present age subtracted from half of Mary’s age at the time she was half as old as Ann. I would not solve this problem for any one but a constant reader.

The Washington Times of 1903-10-27 prints an anonymous poet’s take:

I used to be contented and took comfort in my life,
I dwelt in deep placidity, removed from care and strife,
I never worried as I jogged along my even way,
I worked a bit, I slept a bit, and still had time for play;
I never had such things as nerves; my brain was clear and cool,
But now I am a nervous wreck, and soon will be a fool.

It came about in this wise: One fell day a genial friend,
Whose mind is full of whist and chess and problems without end,
Propounded in a guileless way a problem which, to scan,
Seemed simple: All you had to do was find the age of Ann.
Now, Ann is Mary’s sister, which you’ve no doubt heard before,
And Mary— it is impolite, but Mary’s twenty-four.

Now, Mary’s twenty-four, and she is twice as old today
As Ann was when her sister was — that is, of course, to say,
When Mary was — as old as Mary’s sister Ann is now.
Which brings the question up to you: What’s Ann’s age, anyhow?
That’s what my friend propounded, and I answered, at a pinch,
“The problem you propound, sir, is dead easy — it’s a cinch.”

I worked all day upon it, and I also worked all night,
But somehow that confounded thing would never come out right:
At first I figured it that Ann was twelve, then eight, then ten,
And then I twisted it around and worked it out again.
This time I learned, to my dismay, that figures disagree:
That although Mary’s twenty-four, her sister’s sixty-three!

I raved and swore; my hair I tore; I vented all my spleen
Upon some luckless wight who said that Ann was sweet sixteen.
I’ve worried nights, I’ve worried days, I cannot eat my meals;
I jabber those confounded words until my poor brain reels;
If this thing keeps up many moons, good-bye, old world — farewell!
For me ’twill be the welcome grave, or else the padded cell.

(A less entertaining, but more democratic, sampling of “Ann” correspondence appears in the Evening World of 1903-11-11.)

The craze was short-lived. By 1903-12-21 “Prof. Josh M. A. Long” of the Society for the Prevention of Cruelty to Humor was advertising in the World for any information as to “the whereabouts of ‘How Old Is Ann?’ Has been missing about ten days. Recently released from the Old Jokes’ Home, last heard of in Harlem.”

Footnote: Or in the New York Press of 1903-10-12, signed “John Mahon,” if you believe a column from 1937 (Bellefonte (Penn.) Democratic-Watchman 1937-11-26; South Hill (Va.) Enterprise 1937-12-23). Or maybe the Press of 1903-10-16 (Highland (Va.) Recorder 1951-10-05).

But the true story, as relayed by William J. Lampton in a Scrap-Book article titled “The Vicissitudes of a Problem” (August 1906), is that “How old is Ann?” first appeared in the Newark Evening News of 1903-04-21, headlined “Stories of the Town.” The author was one Robert D. Towne; and when the meme didn’t catch at first in Newark, he tried again, Lampton says, by sending it to the Philadelphia North American and signing it “Perkin Warbeck.” Towne certainly let it be known that he was the “Ann” man (1, 2, 3), earning his obituary a spot among Time magazine’s “Milestones” of 1952-03-03.

Now hear this, from the Sequachee Valley (Tenn.) News of 1903-10-29:

A man is twice the age his wife was when he was the age she is now. When she reaches his present age, their combined years will be 100. Find the age of each.

See also:

• “What is 8÷2(2+2)?” (2019-08-01)

On December 11, 2022, Eric [Albert] suffered a hemorrhagic stroke of his cerebellum. He’s lucky to be alive: around 50 percent of patients die in a month, half of those within the first two days. When we first spoke in April 2023, I immediately asked if his language had been affected, then regretted it. But he was fine, he said. The neurologists don’t know why the stroke happened, or how he recovered his cognition so quickly. He says he repeated the French he was learning to himself in the ICU. I imagined him, was almost ashamed to imagine him, saying ciré ciré ciré, épée épée épée — as if the accent marks were stray neurons in need of repair — or an item from James Joyce’s Finnegans Wake I’d just seen in Word Ways because it’s the only “word” with five or more Q’s: Quoiquoiquoiquoiquoiquoiquoiq! In Joyce’s Wake, it’s preceded by the line: “He lifts the lifewand and the dumb speak.” I remembered analyses of Broca’s aphasia I’d read studying neuroscience in college […]

Page 163:

In P.G. Wodehouse’s 1957 novel, Something Fishy (called The Butler Did It in its American release), Lord Uffenham, every bit an imperious sneeze as his name suggests, barks at his wittier niece, Jane: “[…] Run and ask Keggs what the dickens ‘Adventurer goes in for outrageous road-speed’ is supposed to signify. Tell him it’s urgent. And I want some more coffee… Yer can’t drink too much coffee. It bucks you up. It stimulates the mental processes.” The answer — DESPERADO, an “outrageous” anagram of ROAD SPEED and synonym of “adventurer” — is supplied by the august valet Augustus Keggs. Then Uffenham proceeds to a second stinker: “Desperado, eh? Capital. Now, go and ask him what the devil ‘So the subordinate professional on trial gets wages in advance not without demur’ means.” For the intellectual or the upper-crust peer, the crossword and coffee must be consumed; who labors for it is immaterial. Eliot’s brooder Prufrock “measured out [his] life in coffee spoons,” Margaret Farrar called the Times Saturday crossword a “two cups of coffee puzzle,” Lord Uffenham has his niece make the coffee and his butler solve the puzzle.

The butler did it; but who is there to say, besides Jane, that Uffenham didn’t solve it himself? One imagines Keggs following Uffenham on a summery lawn the color of turtle shells, moving the lord’s croquet balls, muttering “Very good shot, sir” as he does._ […]

Does one, though? Anyway, if you are excited by the mere idea of 300 pages of this sort of thing — accent marks as damaged neurons, lawns the color of not just one turtle shell (i.e. a mottled brown) but several, the names of Simon & Schuster inflated into “fat illegible balloons pricked into oblivion by a skyscraper’s spire” (page 50), a missed encounter between Vladimir Nabokov and his future wife Véra described as “a word golf of only one, as if embalming a simpler past” (page 169), etc., then you’re in for a treat. I myself alternately slogged and skimmed. Many of the book’s chapters were first published individually in magazines, which explains how the same anecdotes keep recurring. “Three-letter word for a female swan” appears on page 54 and again on page 222. There are two capsule biographies of Elizabeth Gorski and at least three of Arthur Wynne.

Last’s shotgun approach does give starting points for some interesting web sleuthing. For example, he reprints a butterfly-shaped Russian-language crossword drawn by Nabokov on 1926-07-06. The image he uses just comes from Wikipedia, which got it from a tweet; but Nabokov’s tiny handwriting is solvably transcribed in Письма к Вере (“Letters to Véra,” 2014). (Thanks to Russian StackExchange for that link!) The solution to Nabokov’s butterfly crossword is (spoiler) here.

Last mentions that Georges Perec, of La Disparition lipogram fame, also wrote crosswords (in French). See one of Perec’s, and solution, at Derek Harrison’s blog. Perec’s crosswords seem to be collected in two books: Les mots croisés (1979; ISBN 2863740032) and Les mots croisés II (1986; ISBN 2867440661).

This reminds me that Stephen Sondheim, of Broadway fame, not only wrote cryptic crosswords for New York magazine in 1968–1969 but is thereby credited with introducing the cryptic to American audiences. His fantastic series of cryptics is not hard to find online, e.g. here and here and even on nymag.com (1, 2, 3). My favorite is 1968-04-29’s “One Shy.”

Fun fact: Sondheim apparently inspired the “Andrew Wyke” character in Sleuth. (Run, don’t walk, to see the 1972 film; skip the bizarre 2007 remake.) From the New York Times of 1996-03-10:

SONDHEIM: Did I ever tell you about the time I met Sir Laurence Olivier? [He] said, “Oh, Mr. Sondheim, I’m so delighted to meet you. I’ve been playing you.” Of course I thought he meant playing records … I said, “Which show?” And he replied, “No, I’ve been playing you in the movie.”

I should mention that a really great book about crosswords and crosswording is Coral Amende’s The Crossword Obsession (2001). It consists almost entirely of crossword constructors’ own words, and is fantastically entertaining. This anecdote from John Samson has lived rent-free in my head for the past 25 years:

SAMSON: The first crossword I sent [NYT puzzle editor Eugene Maleska] was so bad he wrote back: “This is really amateurish. Do us both a favor and never send me another one again. One of your entries, OX MORGUE, clued as ‘Where dead oxen are carted off to,’ is ridiculous; also, your diagram is not symmetric, and on top of that it contains unkeyed letters.” Needless to say, I noticed he had taken the time to reply to me (written in longhand), which prompted another effort on my part. …

By the way, Wodehouse never gives the answer to Lord Uffenham’s cryptic clue “So the subordinate professional on trial gets wages in advance not without demur,” but I see a strong case for PAY UNDER PROTEST.

See also:

• “How old is Ann?” (2026-07-29)

P2548 “copyable_function” (Hava 2023) contains this guidance for implementors:

It is recommended that implementors do not perform additional allocations when converting from a copyable_function instantiation to a compatible move_only_function instantiation, but this is left as quality-of-implementation.

Note that converting in the other direction — from move_only_function to copyable_function — is disallowed: copyable_function can hold only copyable callables, and move_only_function is not copyable.

But copyable_function is interconvertible — both directions — with std::function! Both of these type-erased wrappers are constructible from anything that is copyable and callable, and themselves are copyable and callable. So we can do this:

std::function<int()> f = []{ return 5; };
std::copyable_function<int() const> cf = std::move(f);
std::function<int()> g = std::move(cf);

You’d hope the move from f into cf would be implemented something like this:

and vice versa for the move back into g:

But if copyable_function doesn’t know about function, it might just treat f the same as any old rvalue of copyable, callable type: it might move a copy of f onto the heap to own.

And vice versa for the move back into g: function might just treat cf the same as any old rvalue of copyable, callable type, and move a copy of cf onto the heap to own.

This move operation is still relatively performant — it does only O(1) work for each construction — but after a sequence of \(n\) such constructions, we’ve produced a sort of “linked list” of callables of length \(n\). Each time we invoke g, g’s operator() will call the operator() of its controlled copyable_function, which calls the operator() of its controlled function, which calls the operator() of its controlled lambda (the green object in these diagrams).

As of this writing (July 2026), libstdc++ is the only one of the Big Three STL vendors to have implemented C++26 copyable_function; and as of this writing, they’ve implemented the “bad” behavior diagrammed above.

Consider the following contrived scenario: We have some kind of “handler callback” stored in a std::function. Periodically we consider whether to modify that handler based on user input, e.g.

using Handler = std::function<bool(char)>;

Handler update(Handler h, bool negate) {
  if (negate) {
    h = [f = std::move(h)](char c) { return !f(c); };
  }
  return h;
}

~~~~
Handler h = original_handler;
while (~~~~) {
  ~~~~
  h = update(std::move(h), (input == '!'));
}

Observe that h is moved into update, and moved out again (thanks to implicit move). This uses the move constructor of std::function, which, like most move constructors, is quick and non-allocating.

But suppose one of our coworkers updates part of this code to C++26, where we might reasonably prefer to use std::copyable_function in place of std::function.^[1] If our coworker updates the whole codebase to use copyable_function consistently, there’s no problem. But what if they update only their part, so that their new copyable_function-based code must interoperate with everyone else’s old function-based code? Then we get something like this:

using OldHandler = std::function<bool(char)>;
using NewHandler = std::copyable_function<bool(char)>;

OldHandler update(OldHandler h, bool negate);
  // unchanged, for the benefit of pre-C++26 callers

~~~~
NewHandler h = original_handler;
while (~~~~) {
  ~~~~
  h = update(std::move(h), (input == '!'));
}

This triggers that “linked list” scenario, and h’s call operator gets slower with every round-trip through update — even when negate is false!

I wrote up this little benchmark (Godbolt):

#include <chrono>
#include <cstdio>
#include <functional>

size_t now() {
  static const auto epoch = std::chrono::high_resolution_clock::now();
  auto tp = std::chrono::high_resolution_clock::now();
  return std::chrono::duration_cast<std::chrono::microseconds>(tp - epoch).count();
}

void print_elapsed_time(int i, size_t call_started) {
  printf("Invocation on the %dth iteration took %zu us\n", i, now() - call_started);
}

using OldHandler = std::function<void(int, size_t)>;
using NewHandler = std::copyable_function<void(int, size_t) const>;

OldHandler update(OldHandler h, bool negate) {
  if (negate) { /* do something here, but for this example we don't care what */ }
  return h;
}

int main() {
  NewHandler handler = print_elapsed_time;
  for (int i = 0; i < 160'000; ++i) {
    handler = update(std::move(handler), false);
    if (i % 10'000 == 0) {
      handler(i, now());
    }
  }
}

Run this program on libstdc++ (as of GCC 16) and you’ll see output something like the following:

Invocation on the 0th iteration took 0 us
Invocation on the 10000th iteration took 180 us
Invocation on the 20000th iteration took 296 us
Invocation on the 30000th iteration took 365 us
[...]
Invocation on the 130000th iteration took 1057 us
Invocation on the 140000th iteration took 1108 us
Invocation on the 150000th iteration took 1202 us

Each time we invoke handler(i, now()) the “linked list of callables” hanging off handler is 10,000 elements longer than before, and takes about 100 microseconds longer to chase those pointers. Of course it also holds onto many kilobytes more memory than needed, too. This is a “logical memory leak”: that memory is technically still accessible and will be freed by h’s destructor, but as long as h lives we’ll see our RAM usage creep higher and higher over time, without any change in the “logical” state of our program.

As of July 2026, Microsoft STL makes move_only_function and function interoperate without such “logical memory leaks,” and I bet their copyable_function will also avoid them, once they implement that type. libstdc++’s implementation may improve in the future. Meanwhile, libc++ hasn’t implemented either move_only_function or copyable_function yet, so they still have plenty of time to think about this issue.

Beware of a simpler version of this pitfall, by the way, when writing your own type-erased types. Make sure that when you copy a Printable-holding-an-int, you get another Printable-holding-an-int and not a Printable-holding-a-Printable-holding-an-int. Write unit tests to make sure that still holds when constructing from a Printable&; from a const Printable&; from a Printable&&; and from a const Printable&&. Solving this problem gets more painful if, like the STL, you have more than one type-erased type that need to play well together. Try to avoid getting into that situation if you can; but be aware of this pitfall if you must.

Footnote: Should you abandon function for copyable_function in C++26? Some would say yes. Personally I would say that you ought to abandon both, and write your own type-erased callable instead. It takes less than 100 lines!

In completely green-field C++26 code, sure, I’d recommend copyable_function over function (once your vendor implements it). I would not recommend mixing both types in the same brown-field codebase; I’d pick one and stick with it, while working toward a point where you could update the entire codebase at once.

Well, I wrote to the British Library and got the relevant folios digitized; so I can now present a rough transcription of the Egerton manuscript’s venereal passages. (Sadly there is insufficient communication between the Library’s division who scan things for money and the division who publish things in the digital online catalog; I have the scan, but the British Library’s online catalog still will not. Therefore I’ve uploaded it also to the Internet Archive, where — at least for now — you also may view it.)

For brief inventories of the whole contents of Egerton MS 1995, see:

• The British Museum’s catalog entry

• Item 2105 in an 1865 auction catalog of “highly important and valuable books and manuscripts of a nobleman of great literary and artistic taste”

• James Gairdner, The historical collections of a citizen of London in the fifteenth century (Camden Society, 1876)

Do you have a correction or helpful comment on this transcription? Send me an email!

(Fols. 55v ff.)

Note ye the properteys that longythe to a yonge gentylle man to have knowynge of ſuche thyngys that longythe unto hym that he fayle not in hys propyr termys that longythe unto hym as hyt ſhalle folowe hereynne wrytynge ❦

The Condyſcyons of a grehounde ande of hys propyrnyng ~ propyrteys ~

Thy grehounde moſte be heddyd lyke a ſnake
  y-neckyd lyke a drake
  y-breſtyde lyke a lyon
  y-fotyd lyke a noymon
  y-ſyded lyke a noynon
  y-fotyde lyke a catte
  y-taylyd lyke a ratte

Thenne ys the grehounde welle y-ſhapte ~

Termys of venery ❦

Here begynnythe the Termys of kervynge of Fowlys ande of Fyſchyſſe ❦

Alle the Termys that longgythe unto fyſche ~

Explicit ❦

There follows a poem, “Litylle chyldrynne here may ye lere,” which is transcribed in F. J. Furnivall’s Early English Meals and Manners (Early English Text Society #32, 1868). Picking up again at folio 60r:

Here endythe the boke of Curteſy that ys fulle neceſſary unto yonge chyldryn that muſte nedys lerne the maner of curteſy ~

Explicit; Amen.

There was a man that hadde nought,
There come thevys & robbyd hỹ & toke nought,
He ranne owte and cryde nought,
Why ſhulde he crye, he loſte nought,
Here ys a tale of ryght noughte.

Cronica ſi pences ſi certant oxonienſes
Poſt paucos menſes pugnabũt & angliginences

A “Leonine” witticism also attested elsewhere: “The chronicles tell us that when Oxford men contend / in months all England wars from end to end.”

There follows (fols. 60v–61v) a 2.5-page treatise on prophesying by the first windy night after Christmas: Si ventus fuerit in prima nocte Natalis dm̃ in illo anno Reges et Principes Pontifices et ceteri duutes^(?) vel potentes per univerſum orbem ꝑibunt. Secunda die vel in nocte Natalis dm̃ ventus fuerit unia per univerſum orbem frumenta et grana deficient… For an 11th-century variation on this text, see BnF NAL 873, fol. 200r ff.

There follows (fols. 61v–62v) some Latin commentary on the twelve months.

Here benne alle the namys of the byſcheppe richys of alle yngelonde and thoo that longythe unto Yngelonde ~

Canntyrbury, Yorke, Exceter, Bathe, Salyſbury, Wyncheſtye, Worceter, Rocheſter, Cheſter, Lyncolle, Lonndon, Chicheſter, Ely, Syndavy,^(?) Synt Aſfe, Bangor, Herforde, Cardeaffe, Northewiche, Carlylle, Dyrham [Later addition in a blacker ink: Peterborow] — Non plura^(?)
Smⁱⁱ anñ XXIV^(?)

Deo gracias &^(?) Hylle ~

(“Syndavy” might be St David’s, in Wales; I’m surer that “Synt Aſfe” is St Asaph’s. I can’t hazard a guess on the Latin abbreviations, except to note that if “annum 24” is a regnal year, the 24th year of Henry VI’s reign would have been 1448.)

(Fols. 63r ff.)

For a yong gentylle man to knowe the termys of venery and the crafte whyche the iiij beſtys of venery ~ The herte, the hare, the wolfe, and the wylde Bore — Of venery there bynne nomore.

The Beſtys of the Chaſſe as hyt apperythe ~ The Bucke, the Doo, the Roobucke, the Roodoo, the Bere,^(?) and the rayne dere.

The Elke that ys to ſay a herte of II wyntyr age. ¶ The ſpycurde that ys to ſay a herte of III wyntyr of age.

The Fulmer,^(?) the Fechewe,^(?) the Catte of the mountayne, the Gray, the Fox, the Weſylle, the martoun, the ſquyrelle, the Ottyr, the Polkatte — Alle thes benne of the ſtynkyng Fute or ſtynkying Chaſſe, for the wylle bothe clym and crepe as for the ſubſtance of hem ❦

There ben Grehoundys for game and many othyr dyvers houndys ſum^(?) honeſte & ſum profytabylle as hit ſhalle folowe here in wrytynge ❦

Fyrste a Grehounde, a Baſtarde grehounde, a mongrelle, a maſtowe,^(?) aloundys, lymoire — Alle thes ben take in to the leſſe. ¶ Spaynellys, Rangys, Kenellys, Trerors — Alle thes wylle range for to rayſſe uppe the game. ¶ Bocherhoundys, Dunghylle Currys, and ſmalle Popys for ladys Chambers.

Aloundys ben cuſte, lymors drawyn, a Blode hounde or a dogge for the bowe drawythe, kenettys rennyn, ſpaynellys retryvyn. The Terer gothe undyr the gounde for to feſtyn the Fox or the Graye ❦

A hunt caſtythe of a copylle of Alundys unto an herde of hertys. Then he browythe comfortỹg hys houndys. ¶ The herte takythe the water then the herte ſevſythe^(?) [perhaps fuite? cf. Hare’s The Language of Field Sports p16]. Whenne the he ſwỹmythe downe thanne he fletythe. ¶ Whenne that he ſwỹmythe up on the water thenne he brekythe, or brethythe. ¶ When that he ſwỹmythe ovyr thenne he croſſythe. ¶ Whenne he retornythe thenne he croſſythe.^(?) ¶ Whenne that he takythe dry londe thenne he fleythe. ❦

Explicit.

The namys of hawkys and to what maner of Perſonys that they longe unto every man aftyr hys owne degre and ordyr ❦

There longyn unto an Emꝑour thre maner of hawkys ~ That ys to ſay: an Egylle, a vature, a melounde.

And the ſympylyſte of thes wylle ſlee an hynde Calfe by hym ſelfe, or a doys Fawyn, or a Roo, or a crane or a Buſtarde other wyſe namyde a derefoule. Or a ſtorke or a Fox.

¶ And thes hawkys benne not namyd or callyd unto the lewre; the cauſe ys for they ben ovyr hevy & ponderos for any man to bere. ¶ But they ben made unto the Perke portatyf and ther by they benne governyde.

¶ Ther ben hawkys of the Towre ❦

A Gerfaukyn, a Terſelle, a Grfaukyn — for a kynge; a Faukyn Gentylle and a Taſſelle gentylle — For a Prynce; a Faukyn of the Rooke — For a Duke; a Faukyn Peragryne — for an Erle; a Baſtarde — for every lorde; a Saker and a ſakerette — for a knyght; a laner or a lanerette — for a ſquyer; a Merlyon — for a lady; a Hobby — for a yong ſquyer. Alle thes ben hawkys of the Towre ~ And alle thes fleyn fro the lewre.

¶ The Goſhawke ys for a yeman; a ſperhawk for a preſte; a muſkette for a clerke. Ande thes ben made unto the fyſte ande fle w^t hyr fre wylle yf they benne in ther proſpyte for to make game unto hyr mayſter more þen for any gredynys of mete for they ben noo Rayvyners. ¶ And ſum of the hawkys a fore ſayde wylle be made unto the fyſte welle y-nowe, but they moſte gave grete labyr to bryng them there to. ❦

Explicit.

Gret huntyng by ryvers and wode
Makythe a man ys here to growe thorowe hys hoode.

There follows (fols. 65r ff.) “a nobylle tretys of medyſyns for mannys body.”

Three lost songs from Iolanthe. In April 2026, Marc Shepherd was looking at a manuscript full score of Gilbert & Sullivan’s Iolanthe in the British Library (MS Mus. 1824/1/7/4) and discovered at the back of the manuscript the orchestra parts for these three songs, all cut from Iolanthe before the end of its run and up to now believed about as lost as Thespis:

• “On you they’d set a coronet” (Phyllis, Tolloller, Mountararat, Lord Chancellor)
• “My love for him is dead” (Phyllis)
• “De Belville was regarded as the Crichton of his age” (Mountararat)

More information can be found in Shepherd’s transcription of the full score; in his blog post “Three Deleted Songs from Iolanthe” (May 2026); and in a two-hour video interview with William Remmers, the music director of New York’s Utopia Opera. Utopia Opera has already put recordings of all three songs on YouTube.

Shepherd remarks that Phyllis is a soprano without an aria; the deleted song would naturally have been her big number. But Act II already has its share of wishy-washy love songs — “You plead in vain,” “Oh foolish fay” — and “My love for him” must have seemed too samey. Mountararat, meanwhile, is coming straight from his three-verse comic song “When Britain really ruled the waves”; it doesn’t make much sense to give him a second three-verse comic song in the same act! Still, Gilbert liked “De Belville” per se so much that he republished its words in Songs of a Savoyard (1890); it was only Sullivan’s melody that was lost.

Shepherd is also the coauthor of The Variorum Gilbert & Sullivan, Volume 1 (2015) and has edited the full scores of many of the operettas. The most interesting parts of the Utopia interview, for me, were in the first 15 minutes where Shepherd talks about his ongoing projects (including volumes 2–5 of the Variorum); and circa an hour in, where they discuss the importance of a well-edited score and the crazy things one does when one is lacking. The former reminds me that I have plenty of unfinished projects of my own. For example, after I wrote “In search of Adventure ]I[” (2023-01-09) I did actually receive a playable version, which I haven’t gotten around to blogging about yet; nor have I yet written a post about my online collection of John Philip Sousa operettas. The latter reminded me of what George Bernard Shaw said in the third edition of The Perfect Wagnerite:

It is possible to learn more of the world by producing a single opera, or even conducting a single orchestral rehearsal, than by ten years reading in the library of the British Museum.

Yet another proposed decipherment of Linear A. Via Hacker News: “Amateur linguist” Tom di Mino proposes that the Linear A script of Minoan Crete encodes a spoken language from the Semitic family, and claims to have a list of phonetic values that make sense of at least one inscription. He announced the news via LinkedIn and via Stephen Kosloff’s AI-themed blog, which are weird venues: weird enough to attract attention from hackers with Gell-Mann amnesia, for sure.

I hadn’t yet heard this news when I posted “Katakana and Cypriot” (2026-06-12); that was just serendipity. But on that note I observe that you can find some autodidacts (and even allodidacts) who unironically claim that Linear A encodes spoken Japanese; notably Gretchen E. Leonhardt and Mizuhiro Kuroda. Needless to say, it does not.

Hat sighting in the wild. Last week I visited ArtsWestchester in White Plains, New York, which is housed in a former bank and occasionally uses its armored vault for art installations. This month it’s Rana Amirtahmasebi’s “Continuum: Terra,” in which the artist covers various things (walls, floors, dishes) with the aperiodic monotile known as the “hat.” The hat itself is a recent discovery — it was first discovered in 2022 and announced publicly in March 2023.

ArtsWestchester’s official gallery page inexplicably displays an AI-generated image of some bullshit, rather than Amirtahmasebi’s actual work. What, does nobody at ArtsWestchester own a camera? Top: ArtsWestchester’s AI slop. Middle and bottom: The real-life vault, with hat pattern.

Now, the hat will tile the plane, but you’ll have trouble tiling a curved surface (such as Amirtahmasebi’s lumpy pottery). Her solution was in some cases to deform or truncate the hat, and in other cases to just give up and stop tiling. It wasn’t mathematically satisfying at all. But I liked seeing the hat in the wild, anyway.

See also:

• “Two-Minute Iolanthe” (2026-05-08)
• “Escheresque parquet deformations of an aperiodic monotile” (2023-03-30)

Whoever studies the history of writing, ancient or modern, should be mindful of one basic methodological principle: a comparison between the signs encountered in different scripts should always be treated with skepticism and subjected to severe criticism. There are plenty of examples of signs that are graphically the same, and indeed have the same phonetic value, in scripts that have absolutely nothing in common. A case in point are the nine signs in the classical Cypriot syllabic script, in use on Cyprus until the fourth century BC, and modern Japanese. The signs YE [sic], VE [sic], RO, SE, TO, KO, RU, ME, and PE are written in exactly the same way in both, without anyone ever even imagining an affinity between the script of ancient Cyprus and the kana script of Japan.

These days both katakana and the Cypriot syllabary are encoded in Unicode, so it’s convenient to present the following table of glyphs that (according to their Unicode character names) represent the “same” phonetic values:

See also a nice diagram of a few of these on page 11 of Francesco Soldani’s doctoral thesis “Interconnessione grafica tra i vari sillabari egei e loro leggibilità” (University of Milan, 2013).

The STL sometimes tries to pretend we have “universal template parameters” by overloading templates with the same name. For example, we have both std::get<T>(tp) taking a type and std::get<I>(tp) taking an integer; but these are just two completely different signatures of std::get in the same overload set. Likewise in C++23 you can write either rg | ranges::to<std::vector<int>>() (where the argument to to is a type) or rg | ranges::to<std::vector>() (where the argument is a class template). Again this is accomplished with an overload set — roughly like this (modulo I’ve totally mangled the actual job of ranges::to, which is generally not to forward its arguments to To’s constructor):

template<class To, class... Args>  // #1
auto rangish_to(Args&&... args) {
  return To(std::forward<Args>(args)...);
}
template<template<class...> class To, class... Args>  // #2
auto rangish_to(Args&&... args) {
  return To(std::forward<Args>(args)...);
}

int *a, *b;
auto x = rangish_to<std::vector<int>>(a, b);      // OK, #1
auto y = rangish_to<std::vector>(a, b);           // OK, #2

But even this overload set won’t accept a caller like:

auto z = rangish_to<std::ranges::subrange>(a, b); // error

because std::ranges::subrange as a template argument matches neither class To nor template<class...> class To. It’s actually a class template of two type parameters and a constant! (cppreference.)

enum class subrange_kind : bool {
  unsized,
  sized,
};
template<
  input_or_output_iterator I,
  sentinel_for<I> S = I,
  subrange_kind K = sized_sentinel_for<S, I> ? sized : unsized>
class subrange { ~~~~ };

We could accept subrange ad-hoc via a third overload:

template<template<class,class,auto> class To, class... Args>  // #3
auto rangish_to(Args&&... args) {
  return To(std::forward<Args>(args)...);
}

but in general there’s no way to write out the set of all possible kinds of class templates, so we can’t make our rangish_to accept all of them.

You might point out that the real ranges::to wouldn’t accept subrange anyway, because the real ranges::to accepts only containers like vector, not views like subrange or span. However, this deficiency is likely to be DR’ed in the near future thanks to Hewill Kang’s well-received P3544 “ranges::to<view>.”

C++26 Reflection to the rescue! C++26 Reflection is “untyped”: it uses a single type std::meta::info to represent the reflections of every kind of thing in the language. Therefore a template parameter of type meta::info alone can represent std::vector, std::ranges::subrange, std::array, or anything else (std::vector<int>, size_t, 42, std::ignore…) and so we can write a single function template like this (Godbolt):

template<std::meta::info To, class... Args>
auto meta_to(Args&&... args) {
  return typename [:To:](std::forward<Args>(args)...);
}

auto x = meta_to<^^std::vector<int>>(a, b);      // OK
auto y = meta_to<^^std::vector>(a, b);           // OK
auto z = meta_to<^^std::ranges::subrange>(a, b); // OK!

For the cost of two characters ^^ (and a really cryptic error message if you forget the typename keyword), C++26 seems to have given us universal template parameters!

(Does this mean P2989 “Universal template parameters” is obsolete? IMHO, yes. But I can see how one might debate it: those two ^^ characters are kind of ugly.)