Carnival of Mathematics #34

Welcome to the 34^th Carnival of Mathematics!

There are 34 musicians who have performed as members of Deep Purple or Rainbow [Ritchie Blackmore is notorious for his personnel changes.]. And on the subject of music, Maria Andersen from the Teaching College Math Technology Blog submitted the song “How Do you Write Your Math in Online Classes” (sung to the tune of “How do you solve a problem like Maria”), complete with Listen Now.

There are 34 islands in the Mediterranean (according to Wikipedia). One of the islands is Samos, where Pythagoras was born. And in An Ancient Mathematical Crisis, Denise of Let’s Play Math! explains how the proof that √(2) is irrational shook the philosophical world of the Pythagorean Brotherhood.

Two possibly interesting facts about the number 34 are that 34 is the 4th root of 10⁶+(10³+1)·336, and that 34 is equal to 3²+5², which makes it the first composite number that is the sum of two nonunit coprime squares. Two absolutely interesting posts were submitted by Julie Rehmeyer from Math Trek: Detangling DNA, which examines the knots that form in DNA, and Insights into Symmetry, which explains a bit more about the math behind this year’s winners of the Abel Prize, awarded this month.

The number 34 is a Fibonacci number. It’s possible to use a 2×2 matrix (and eigenvalues and diagonalization) to find an explicit formula for the Fibonacci numbers. Another thing that can be done with a 2×2 matrix is to take its determinant, and in Determinants as proportions, on Matt-a-matical Thinking, Matt writes about how determinants can be used to solve proportions and why it might be worthwhile to teach proportions that way.

The number 34 can be written as 1/4 + 2/4 + 3/4 + 1 + 5/4 + … + 4 [the sum of all the consecutive quarters up to 4]. To explore consecutive integers rather than consecutive quarters, head over to Math and Logic Play: Adding Up a Consecutive Range of Integers, shared by Praveen at Math and Logic Play.

The number 34 is the smallest natural number N where N-1, N, and N+1 have the same number of factors. The numbers 0 and 1 don’t have the same number of factors, but within the interval [0,1] it’s possible to pick some random numbers and look at the largest one; Barry poses some intriguing questions about this in Random Expectations, posted at fashionablemathematician – mathematics.

Country star Gretchen Wilson just completed her GED at the age of 34. (Contratulations Gretchen!) She had dropped out of school in 9^th grade to start work, but if she’d stayed in school her teachers might have given quizzes occasionally. And I’m sure that those teachers would have appreciated having a site to go to for creating tests, as Larry Ferlazzo shares “That Quiz” posted at Larry Ferlazzo’s Websites Of The Day For Teaching ELL, ESL, & EFL.

There were 34 States in the US at the start of the American Civil War. This kind of information is often represented with a map, which is a kind of graphic. And if you’re going to include graphics in any sort of presentation, you would do well to make them accurate and not slyly misrepresent the data. jd2718 shares this exact kind of situation in Lying with graphics at NYCDOE Central.

On average, the amount of time that passes between two people dying of heart diseases in the US is 34 seconds. Even if you didn’t know that (you didn’t know that, did you?) if someone told you the average was -34 seconds, or 10 billion trillion seconds, you’d probably know enough to question their results. A reality check, in other words. Julie Bloss Kelsey, aka Mama Joules, shares her own mathematical anecdote about reality checks and saving money in Always get a reality check.

One last fact is that the STS-34 (the Shuttle Atlantis) took the Galileo space probe into orbit in October 1989; Galileo arrived at Jupiter in 1995, and remained in orbit gathering data on the Jovian system until 2003. And one last contribution is that today, May 30, is the birthday of Karl Feuerbach, as TwoPi shares in Mathematician of the Week: Karl Feuerbach, the third of a new weekly (Sunday) post here on 360.

See you at the next Carnival of Mathematics, on Friday the 13th of June, over at CatSynth!

Hoorah for the Hexadecagon!

There was a Hexadecagon in the New York Times Wednesday, in an article about glassblowing. First the article talked about people blowing glass at the Corning Museum GlassLab:

Except the glassblowers weren’t from 1955. Then the article explored the creation of glass designs, including knit glass (!!!) and New York City inspired pretzels. And, as promised in the title, there was also a hexadecagon. It looked a little like this

Except in wasn’t in Las Vegas. And it had 16 sides.

With that inspiration, I looked up hexadecagon to see what I could learn about it. I found that a hexadecagon is also known as a hexakaidecagon, and can be constructed with a straightedge and compass. And I found that origami madness made a neat origami design with four hexadecagons:

Photo by origami madness. Some rights reserved.

Here it is unfolded:

Photo by origami madness. Some rights reserved.

And oschene made some Fujimoto cubes with hexadecagonal irises (irides?)

Photo by oschene. Some rights reserved.

Photo by oschene. Some rights reserved.

And finally, the Imperial Seal of Japan (Crest of Chrysanthemum) isn’t quite a polygon, but if it were then it would be a hexadecagon.

Licensed under Creative Commons Attribution ShareAlike

Carnival Submissions Welcome!

Just a reminder that in two days (Friday May 30), we’ll be hosting the 34^th Carnival of Mathematics!!  Submissions are welcome via the Official Submit Form, via the comments below, and via email to hlewis5 followed by the @ and then ending with naz.edu (put something like “Carnival” in the subject line if you think of it).

In theory we’ll be putting it together tonight and tomorrow.  In reality, we’ll be working on it tomorrow night and Friday morning, so anything submitted anytime Thursday will certainly be included.

Polygons on Mars

The Phoenix Mars Lander arrived safely on Mars Sunday night! This is a particularly big deal because the previous Lander didn’t: the Mars Polar Lander was due to arrive on Mars in December 1999 but, for still-unknown reasons, communications stopped suddenly about 6 minutes before it was due to enter the Martian atmosphere and its exact whereabout remain unknown. (Incidentally, the Polar Lander was part of the Mars Surveyor mission. The other part was the Mars Climate Orbiter, which ran into a little $125 million problem of its own when the teams didn’t translate between imperial and metric units.)

But back to the Phoenix Lander: the Phoenix had no trouble, landed perfectly, and is already sending back pictures. Our newspaper this morning showed the following photo of polygons on Mars:

The newspaper made a big deal about the fact that there were polygons and when I looked at the picture my response was along the lines of, “Umm. Okay.” But then I found NASA’s image page and the caption for this picture explains that this polygonal pattern is “similar in appearance to icy ground in the arctic regions of Earth”. So then I went to My Favorite Source and found this photo from Canada’s Northwest Territories (taken by Emma Pike), where the polygons are more noticeable:

The polygons are formed by water getting in cracks, freezing, and then expanding. When it gets cold enough (-17°C, or close to 0°F) the ice contracts rather than expands, and that leaves even bigger cracks (called ice wedges) for more water to get in, etc. So finding polygons on Mars could be a big deal indeed.

There was also another neat NASA picture of the Phoenix Lander landing:

You can see the parachute and everything! According to the the FAQ page at the University of Arizona:

Phoenix is very grateful to the Mars Reconnaissance Orbiter (MRO) team for that otherworldly picture. It was very, very good math. MRO was moving about 3.4 km/sec (30,000 mph). Phoenix, at the time of parachute deployment, was moving between 700-130 mph.

Hooray for very, very good math and for the Phoenix Lander!

Memorial Day

Today is the day in the US for remembering those who died in US wars and conflicts.

Memorial Day has always been in May, although the date has changed over time. It began as a holiday to commemorate those who had died in the Civil War, which ended in 1865. There were several such ceremonies in the year after the war, but the “official” beginning (as declared by Congress and President Lyndon B. Johnson in 1966) was one in Waterloo, New York on May 5, 1866, which honored local veterans who had died in the Civil War.

Two years later, Major General John A. Logan of the Grand Army of the Republic announced what was originally known as Decoration Day.

The 30th day of May, 1868, is designated for the purpose of strewing with flowers, or other decorating the graves of comrades who died in defense of their country during the late rebellion, and whose bodies now lie in almost every city, village and hamlet churchyard in the land. In this observance no form of ceremony is prescribed, but Posts and comrades will, in their own way, arrange such fitting services and testimonials of respect as circumstances may permit.

The first (official) Decoration Day ceremony was held at Arlington National Cemetery.

While the ceremony was originally intended to honor only those who died in the Civil War, after World War I it was changed to honor all who had died in US wars. The date remained May 30 until 1971, when the Uniform Holidays Act changed it to be the last Monday in May.

One interesting note: Frank Woodruff Buckles was honored in a special ceremony this weekend. At age 107, he is the last known US veteran of World War I (having lied about his age in order to enlist in the Army). You can read more about his story here, and more about the history of Memorial Day here.

Mathematician of the Week: Karl Feuerbach

Karl Feuerbach was born on May 30, 1800 in Jena, Germany, into an academic family. His father was a law professor, and five of the eight children earned doctorates, three going on to become professors (notably including Ludwig Feuerbach, whose writings had a significant influence on Karl Marx). Karl Feuerbach’s teaching career was cut short by failing health, and he died two months short of his 34th birthday.

Feuerbach’s most significant result concerned the 9 point circle associated with a triangle. Given any triangle, we can identify the following 9 points: the midpoints of the three sides of the triangle, the feet of the three altitudes of the triangle, and the three midpoints of the line segments connecting the orthocenter with each vertex. In any triangle, these 9 points lie on a circle (i.e. they are co-circular).

Feuerbach showed that this circle is tangent to the incircle of the triangle (the largest inscribed circle), as well as the three excircles (each excircle is tangent to one side of the triangle and the extensions of the other two sides). The point of intersection of the incircle and the 9 point circle is now known as the Feuerbach point of the triangle.

Other mathematicians with significant anniversaries for the week of May 25 through May 31:

May 25: Birthday of Karl Peterson [1828]; death of Johann Radon [1956]

May 26: Birthdays of Abraham de Moivre [1667], Yurii Sokolov [1896], and Otto Neugebauer [1899]

May 27: Death of Arthur Schönflies [1928]

May 28: Birthdays of Jacopo Riccati [1676], Johann Bernoulli (II) [1710]; death of Rolf Nevanlinna [1980]

May 29: Birthday of Finlay Freundlich [1885]; death of Frans van Schooten [1660]

May 30: Birthdays of Karl Feuerbach [1880] and Eugène Catalan [1814]; death of Vladimir Steklov [1926]

May 31: Deaths of Evariste Galois [1832] and George Green [1841]

Source:  MacTutor History of Mathematics Archive

Today’s Art Lesson: Theorem Painting

Theorem Painting is not, as you might think, rendering one’s favorite equations to canvas. Rather, it is a style of painting using stencils (called “theorems”) that may have developed abroad, possibly in England, but was certainly around in the US in the early 1800s. In 1830, Matthew D. Finn published the book Theoremetical System of Painting, in which he explains:

Now, of all the methods ever introduced in the art of painting by water colours, that of the theoremetical takes the lead; more particularly in flower painting, in which the beautiful tints, lights and shades, which can be so easily accomplished, even by a child, remain unrivalled; whilst the mystery of the performance lies hidden from the nicest critic. (quoted here)

The mystery of the performance being that a separate stencil is created for each overlapping shape in the painting, so you don’t have to be especially artistic to create the final picture. According to Jean Hansen, the stencils are “cut in such a manner that no two areas immediately next to each other can be placed on the same stencil. Thus, any theorem will require the sequence of two or more stencils or overlays.” It may be that this following of rules (stencils) is why the word Theorem is used — that’s what various art sites hint at — but in truth I can’t find any sure reason for the connection. The 1971 Compact Edition of the Oxford English Dictionary only gives the mathematical sense of the word; the 1987 Supplement adds the artistic definition, but only defines it as a stencil or design executed by means of a stencil and mentioned that it is also known as Formula Painting.

One final note: The Supplement also quotes the Federal Gazette from April 1, 1834, which refers to “Theorem painting on velvet….” To my mind, this suggests that Theorem Painting is a precursor to that famed artistic piece, the Velvet Elvis.

The book cover is of The Art of Theorem Painting by Linda Carter Lefko and Barbara Knickerbocker (Crafter’s Corner, Inc., 2002), which gives a history of Theorem Painting and also a collection of patterns.

Getting ready for Carnival of Mathematics #34

On Friday May 30, we’ll be hosting the 34^th Carnival of Mathematics!  We’d love to have your submissions:  you can post them directly in the comments here, or send them by email to hlewis5 followed by the @ and then ending with naz.edu (put something like “Carnival” in the subject line if you think of it), or via the Submit form here.

Teacher Arrested

One of our students sent us the following news story to post here. (Thanks Kristen!)

A public school teacher was arrested today at John F. Kennedy International Airport as he attempted to board a flight while in possession of a ruler, a protractor, a set square, a slide rule, and a calculator. At a morning press conference, the Attorney General said he believes the man is a member of the notorious Al-gebra movement. He did not identify the man, who has been charged by the FBI with carrying weapons of math instruction.

“Al-gebra is a problem for us,” the Attorney General said. “They desire solutions by means and extremes, and sometimes go off on tangents in a search of absolute value. They use secret code names like ‘x’ and ‘y’ and refer to themselves as ‘unknowns,’ but we have determined they belong to a common denominator of the axis of medieval with coordinates in every country. As the Greek philanderer Isosceles used to say, “There are 3 sides to every triangle.” When asked to comment on the arrest, President George Bush said, “If God had wanted us to have better Weapons of Math Instruction, He would have given us more fingers…………and toes.”

White House aides told reporters they could not recall a more intelligent or profound statement by the President.

Hexagons in the News: Nanotubes

Nanotubes are back in the news! Nanotubes are sheets of carbon atoms, one atom thick, that roll up to form strong cables for tennis rackets and baseball bats. Very strong cables, in turns out, as in a possible material for a space elevator.

But even if the space elevator doesn’t work out, it turns out that carbon nanotubes can be used to make near-ideal black objects: things which absorb light completely, not refracting any of it. ‘Darkest Ever’ Material Created on BBC News in January explains that this is useful for creating solar cells and solar panels.

So what does does this have to do with mathematics? Quite a bit, if you look on wikipedia or if you actually build anything physical with it. But even at a simpler level, one neat property is that the carbon atoms bind in hexagons. You can see that in the picture below of a hexa-tert-butyl-hexa-peri-hexabenzocoronene. (How’s THAT for a word?)

The hexagons in the picture above are regular hexagons, meaning that all the sides and all the angles are equal. You can completely tile a plane with regular hexagons, without leaving any gaps. And this means that the hexagons in the carbon nanotubes must not be exactly perfectly regular, or else there would be no way for them to roll up. Either the angles or the sides must be a little off.

This irregularity doesn’t actually harm the carbon nanotubes. Unfortunately, they have bigger problems right now: BBC news reported Tuesday that when some fibers got into the lungs of mice, they caused inflammation and legions, like asbestos. Poor nanotubes. That doesn’t sound the death knell, however; it’s only one study, and so more research, more money, etc. etc. have to be dedicated to the topic. But it might be a while before we see that solar-powered space elevator.

Pictures used under GNU Free Documentation License.

Ode to the Impala

It was already 14 years old and had well over 100,000 miles on it. And a big dent in the front, half a grill, and no side mirrors. But it only cost $500, courtesy of our friend Sue, so we considered it a deal.

It broke down in a lot of places. Certainly in Madison, and also in Iowa, Missouri, Nebraska, Arizona, Nevada, California, New York, and Alabama. We had to bypass visiting a good friend in Albuquerque on our way home from the Joint Mathematics Meetings in San Diego (1997) because it turned out our radiator was busted and the only way to keep the car from completely overheating was to blast the heat and to take the southernmost route across the Rockies.

But most of the time it could be fixed easily, and we met quite a few people while we were stranded. There were those hunters when we got stuck by the side of the road, trying to get as far north as possible from Florida while we outraced a storm (they patched us up with duct tape). There was the stranger who let us into his house to use his phone when we got stranded taking my mom and sister to Niagara Falls. Of course, there was also that creepy guy at the rest stop who made a point of restarting his car and driving it all the way to the farthest corner of the lot before turning it off again, so I had to walk half a mile up the highway to find someone to give us a jump, but overall people were friendly and the car’s reliability a harmless quirk.

At least for a few years. The day came when our mechanic refused to fix it: “It’s 16 years old and has gone 180,000 miles. It’s time to say good-bye.” So we got another car, but kept the Impala anyway, though it was down to 6½ cylinders and needed to be warmed up for 15 minutes before moving. Eventually we had a kid and the romantic breaking down in strange places started to seem a lot less romantic, and we sent it onward to the Great Garage in the Sky.

During its last year, the car didn’t have very good gas mileage. Ten miles per gallon was about as far as it could manage, and from a cost perspective that just wasn’t very economical. In 2000, the year the car Went Away, gas had jumped up to $1.50/gallon, which translated into a cost of around 15 cents a mile. At the time, that was expensive.

I was thinking on this today when I was looking at gas prices. Our car now (which does have side mirrors) gets a whopping 26 miles per gallon. Of course, gas around here is $3.90/gallon, so I did a quick check and you know what? That translates into 15 cents a mile. And 15 cents today is even a little less than 15 cents eight years ago, in an “adjust for inflation” kind of way.

Ah, the Impala. It was a car ahead of its time.

The Impala, stranded 99 miles from Barstow.

Blowing out candles with your very last breath

Today is the birthday of Henry White, a mathematician who had tremendous influence on the development of the US mathematical community at the turn of the 20th Century; most notably, he was involved in the leadership of the American Mathematical Society, played a role in the creation of the AMS Colloquium Series, and served as editor of Annals of Mathematics as well as the Transactions of the AMS.

Today is also the anniversary of Henry White’s death.

The MacTutor History of Mathematics website notes that he is the only person in their database whose birth and death dates coincide. As they have thousands of biographies on their site, most of which apply to deceased individuals, and a significant number of which should have reliable birth and death dates, this dearth of birthday-death is a tad surprising.

How likely is one to die on one’s birthday?

Some assumptions:

• We’ll limit ourselves to individuals who reach “maturity”; one expects childhood illnesses and other factors cause the very young to have a radically different distribution of death dates measured relative to birth dates mod 365 than one would find in the adult population.
• We’ll ignore those who are born or who die on a Leap Day.

As a working hypothesis, we’ll assume that one is as likely to die on one’s birthday as on any other day of the year. Under that hypothesis, the probability that one would die on one’s birthday is 1/365.

At this point I would love to be able to go into the details of a hypothesis test, using the data from the MacTutor site as a sample. [It isn’t a random sample, but one might posit the assumption that the bias of choosing famous mathematicians from history will have no impact on the birthday-deathday relationship.] Their data, though, isn’t so easy to compile into a spreadsheet, and I have a class to teach in a few hours….

I did, however, find a different data set on the web: the site Who’s Alive and Who’s Dead posts lists of (mostly 20th C) celebrities (actors, musicians, politicians, and the like) who satisfy the criteria for inclusion that the site uses (essentially: someone has to be famous, and potentially the subject of a “Is so-and-so still living? I haven’t seen them in anything lately.”).  Again, not a representative sample of the public at large, but a sample whose biases [hypothetically] would have no impact on the likelihood of dying on one’s birthday. And most importantly, I was able to capture the data into Excel and hack on it.

The Who’s Alive and Who’s Dead data list birth and death dates for 953 individuals; of those, 4 of them died on their birthday: actors John Banner, Ingrid Bergman, and Mike Douglas, and social activist Betty Friedan. The sample size is still a bit small to apply an hypothesis test, but certainly 4 out of 953 is consistent with the claim that the theoretical probability is 1/365.

Several studies have attempted to show a connection between one’s birth and death dates, for a variety of populations. One published study went so far as to claim that celebrities manage to postpone their deaths until after their birthdays — that celebrities are less likely to die in the month prior to their birthday, and more likely to die in the month afterwards. Certainly one can craft plausible scenarios to explain such a pattern, but doubts have been raised as to whether such a pattern exists at all. (Heather Royer and Gary Smith take on the myth of a “death dip” in an article in Social Biology.)

The data on celebrities from Who’s Alive and Who’s Dead show that 82 of the 953 died in the 30 days prior to their birthday; 84 died within 30 days after their birthday. (One would expect on average 78 or so.) This data is consistent with Royer and Smith’s claim that no such death dip occurs.

Gary Smith’s website includes other articles of interest, including work debunking the notion that Americans of Chinese and Japanese ancestry are more likely to die on the 4th day of a month (putatively because of fears generated by cultural perceptions of 4 as an unlucky number), and a study refuting the claim that people with unpopular names have shorter life expectancies.  Highly recommended reading for the statistically inclined.

The Larsson-Cederlöf Bunch

Here’s the story
Of a quantum code
That was happy as a little code could be
Because all of its parts were so secure
At least they seemed to be.

Here’s the story
Of a man named Larsson^*
Who was thinking a vulnerability could be shown
In collaboration with a former grad student^**
But the results weren’t known

Till that one day when those math guys cracked that code
And they knew that it was much more than a hunch
That this code could be broken by a hacker
Who gave a one-two punch.^***

A one-two punch
A one-two punch
And that code’s security is now questioned a bunch.^****

_________________________________________

^*Jan-Åke Larsson, a math prof at Linköping University in Sweden
^**Jörgen Cederlöf, who now works for Google
^***by “simultaneously manipulating both the quantum-mechanical and the regular communication needed in quantum cryptography” according to Science Daily
^****Or maybe just a little – according to IEEE, Larsson and Cederlöf have provided a patch

Mathematician of the Week: Thoralf Skolem

Thoralf Skolem was born on May 23, 1887 in Sandsvaer Norway. Much of his mathematical research involved the interaction of logic and algebra, including work on lattice theory and set theory. He is probably best remembered in connection with the Lowenheim-Skolem Theorem, and Skolem’s Paradox.

One version of the Lowenheim-Skolem Theorem states that any countable first order theory with an infinite model in fact has models of all infinite cardinalities.

Some examples:

• Groups: The axioms for a group constitute a first order theory. We know there are infinite models for that theory (i.e. we know that infinite groups exist which satisfy the axioms); hence by the Lowenheim-Skolem Theorem, there exist groups of all infinite cardinalities.
• Peano Arithmetic: The Peano axioms for the arithmetic of the natural numbers constitute a first order theory, with the natural numbers themselves constituting an infinite model. It follows that there are uncountable models of this theory as well; in particular, the Peano axioms fail to uniquely characterize the natural numbers, and indeed no first order theory can uniquely characterize N. More generally, no first order theory can uniquely characterize any infinite mathematical object. This fact can be used to prove that a particular property is not a first order property. For example, since the field of the real numbers R is the unique complete linearly ordered field, the notion of completeness cannot be described in terms of first order logic in the language of fields.
• Zermelo-Frankel Set Theory: The Zermelo-Frankel axioms for set theory, the putative foundation for 20th century mathematics, constitute a first order theory. If the ZF axioms are consistent (if the axioms of set theory have an infinite model), then the Lowenheim-Skolem Theorem shows that ZF has a countable model, a universe which is countable, but internally satisfies all of the properties of ZF.

The existence of a countable model of set theory is sometimes referred to as Skolem’s Paradox, for within ZF one can prove the existence of uncountable sets, and any model of ZF would contain sets that from the perspective of the model are uncountable.

How could such a thing occur?

Countability is the property of being able to be placed in one to one correspondence with N. For an “uncountable” set U within a countable model of set theory, we can see externally that the set U must be countable (there exists a bijection between U and N), but the bijection itself is not an object in the model. Hence in the model’s interpretation of set theory, U is uncountable.

Skolem remained active in research up until his death in 1963 at the age of 75.

Other mathematicians with significant anniversaries for the week of May 18 through May 24:

May 18: Birthdays of Oliver Heaviside (1850) and Bertrand Russell (1872)

May 19: Birthday of Serge Lang (1927); death anniversary of Francis Maseres (1824)

May 20: Birthday (1861) and death anniversary (1943) of Henry White

May 21: Birthdays of Albrecht Dürer (1471), Gaspard-Gustave de Coriolis (1792), and Edouard Goursat (1858); death of Ernst Zermelo (1953)

May 22: Birthday of Albrecht Fröhlich (1916)

May 23: Birthday of Thoralf Skolem (1887); death of Augustin-Louis Cauchy (1857)

May 24: Birthday of William Chauvenet (1820); deaths of Nicolaus Copernicus (1543) and Sylvestre Lacroix (1843)

Source for dates and biographical info on Skolem: The MacTutor History of Mathematics Archive.

Carnival of Mathematics #33: Walking Randomly Saves The Day!

The 33^rd Carnival of Mathematics is up at Walking Randomly! (who also hosted the 25^th Carnival). This carnival was homeless until a couple days ago when Walking Randomly stepped up to the plate. For only having a few days to put it together, it contains an impressive number of posts. Many, but not all, have a technology bent.

The Carnival post also explains the following cool fact about the number 33: Most numbers below 1000 that are not of the form 9n±4 can be written as a sum of three signed cubes (meaning the cubes are positive or negative integers). As an aside, the way is not necessarily unique. For example, from this source:
12=7³+10³+(-11)³ and
12=9730705³+(-9019406)³+(-5725013)³.

Up until 1999, the number 30 was the smallest positive integer that wasn’t of the form 9n±4 where there was no known way to write it as a sum of three signed cubes. But less than ten years ago a way was found (see this breakdown of numbers under 100), and now 33 is the smallest such number.