Math and Pancakes
Image: Wikimedia Commons

If you apply one straight cut to a pancake, pretty clearly you’ll get 2 pieces. With two cuts, the most you can get is 4. What’s the greatest number you can produce with three cuts? If the cuts meet neatly in the center, you’ll get 6 pieces, but if you’re artfully sloppy you can make 7 (above). Charmingly, this leads us into the “lazy caterer’s sequence” — the maximum number of pieces you can produce with n straight cuts:

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, …

Generally it turns out that the maximum number for n cuts is given by the formula

\displaystyle p = \frac{n^{2} + n + 2}{2};

each number equals 1 plus a triangular number.

A related question is the pancake flipping problem. You’re presented with a spatula and an untidy stack of pancakes of varying sizes. You can insert the spatula at any point in the stack and flip all the pancakes above it. What’s the least number of flips required to sort the pancakes in order of size? Interestingly, no one has found a general answer. It’s possible to work out the solution for relatively small stacks (in which the number of pancakes is 1, 2, 3, …):

0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, …

But no one has found a formula that will tell how many flips will get the job done for a stack of any given size.

The problem has an interesting pedigree. Bill Gates worked on it at Harvard (PDF), and David X. Cohen, who went on to write for The Simpsons and Futurama, worked on a related problem at Berkeley in which the bottom of each pancake is burnt and the sort must be completed with the burnt sides facing down.

CCNY mathematician Jacob Goodman, who first hit on the pancake flipping problem while sorting folded towels for his wife, submitted it to the American Mathematical Monthly under the name Harry Dweighter (“harried waiter”). His household chores have produced at least one other publication: After some thoughtful work with a swivel-bladed vegetable peeler, he published “On the Largest Convex Polygon Contained in a Non-convex n-gon, Or How to Peel a Potato.”

(Thanks, Urzua.)


Chess master Wilhelm Steinitz was having a heated political argument.

His opponent said, “Do you think you understand politics because you can play chess?”

Steinitz said, “Do you think you understand politics because you can’t play chess?”

Please Help!

I spend hundreds of hours each month researching and writing Futility Closet, and that effort is supported entirely by the readers. If you value this site, please consider making a contribution to help keep it going.

You can make a one-time donation on our Support Us page, or you can pledge a monthly contribution on Patreon to receive bonus posts and other rewards — you choose the amount to contribute, and you can change or cancel it at any time.

Thanks for reading, and thanks for your support!

Edifice Wrecks

giarre, sicily

Between the 1950s and the 2000s, the Sicilian town of Giarre started a series of ambitious building projects as its politicians competed to create jobs and secure funds from the regional government. Unfortunately, there was no need for the buildings — Giarre’s population is only 27,000 — and today the seaside town hosts 25 half-built and abandoned constructions, including an amphitheater, a sports stadium, a polo ground, and a swimming pool.

“Giarre offers the extreme form of a condition found in most cities, making it a parable of urban planning,” writes social geographer Alastair Bonnett in Off the Map. “It is the epicentre not of merely an Italian but a global phenomenon of accreted unfinished visions.”

“Several companies started the projects without the intention of finishing them,” architect Salvo Patane told the BBC. “These were projects started so as not to lose funds that were available from the regional government. More than waste, this was bad politics.”

Community activist Claudia D’Aita wants to reconceive the abandoned constructions as a park — “a kind of open-air museum” — exhibiting a cautionary new architectural subgenre. They would call it the Archaeological Park of Sicilian Incompletion.

Recycling Poetry

pimenta anagram

In 1987, Portuguese poet Alberto Pimenta took the sonnet Transforma-se o amador na cousa amada (The lover becomes the thing he loves), by the 16th-century poet Luís de Camões, and rearranged the letters of each line to produce a new sonnet, Ousa a forma cantor! Mas se da namorada (Dare the form, songster! But if the girlfriend).

Here’s Camões’ (curiously apposite) original poem, translated by Richard Zenith:

The lover becomes the thing he loves
by virtue of much imagining;
since what I long for is already in me,
the act of longing should be enough.
If my soul becomes the beloved,
what more can my body long for?
Only in itself will it find peace,
since my body and soul are linked.
But this pure, fair demigoddess,
who with my soul is in accord
like an accident with its subject,
exists in my mind as a mere idea;
the pure and living love I’m made of
seeks, like simple matter, form.

Carlota Simões and Nuno Coelho of the University of Coimbra calculated that the letters in Camões’ sonnet can be rearranged within their lines in 5.3 × 10312 possible ways.

Interestingly, after Pimenta’s anagramming there were two letters left over, L and C, which are the initials of the original poet, Luís de Camões. “It seems that, in some mysterious and magical way, Luís de Camões came to reclaim the authorship of the second poem as well.”

In 2014, when designer Nuno Coelho challenged his multimedia students to render the transformation, Joana Rodrigues offered this:

Related: In 2005 mathematician Mike Keith devised a scheme to generate 268,435,456 Shakespearean sonnets, each a line-by-line anagram of the others. And see Choice and Fiction.

(Carlota Simões and Nuno Coelho, “Camões, Pimenta and the Improbable Sonnet,” Recreational Mathematics Magazine 1:2 [September 2014], 11-19.)

Tech Talk

In 1944 British graduate student John Hellins Quick published a description of the “turbo-encabulator,” a marvelously sophisticated device whose workings are understandable only by engineers:

The original machine had a base-plate of prefabulated aluminite, surmounted by a malleable logarithmic casing in such a way that the two main spurving bearings were in a direct line with the pentametric fan. The latter consisted simply of six hydrocoptic marzelvanes, so fitted to the ambifacient lunar waneshaft that side fumbling was effectively prevented. The main winding was of the normal lotus-o-delta type placed in panendermic semi-bovoid slots in the stator, every seventh conductor being connected by a nonreversible tremie pipe to the differential girdle spring on the ‘up’ end of the grammeters.

General Electric, Chrysler, and Rockwell Automation have all sung the device’s praises, even if no one can quite explain what it does. Actor Bud Haggart shot the video above in 1977 after completing an industrial training film for General Motors.

The rest of us will just have to take its wonders on faith. When Time magazine published the description in 1946, one reader wrote, “My husband says it sounds like a new motor; I say it sounds like a dictionary that has been struck by lightning.”