Malba Tahan: The exciting conclusion!

A photograph of three camels in EthiopiaLast Wednesday, I continued my series on People Who Are Not What They Seem with Malba Tahan, the Islamic intellectual and writer who was actually the fictional alter ego of a Brazilian math teacher, created to help his students learn word problems.

That post ended on a mathematical cliffhanger (bwahahahaha!). Our heroes, Tahan and his friend Beremiz Samir, happened upon an argument among three brothers about how to divide up their father’s inheritance. Their father left them a herd of 35 camels, with the following instructions (as reported by the eldest brother):

According to the express wishes of my father half of them belong to me, one-third to my brother Hamed, and one-ninth to Harim, the youngest.

Samir’s solution was clever, but it required some risk – he added into the herd the camel that Tahan was riding, making a new herd of 36. He then divided the new herd according to the father’s instructions: one-half (18) to the eldest, one-third (12) to the middle, and one-ninth (4) to the youngest. All three brothers were satisfied with this arrangement, which left two camels remaining. One, of course, was Tahan’s that had been added at the beginning. Samir requested the other as his payment for arranging this solution – and since all three brothers were satisfied, they agreed. Samir grabbed the strongest, most beautiful member of the herd, and the pair rode off together into the sunset.

It’s a happy ending. Everyone is satisfied, especially our heroes. And you have to admire Samir’s Raven-level trickeration in getting something for nothing. But how did he solve the problem?

When faced with a word problem, often the best first step is to write down what you know and what you want to find out. Before Tahan and Samir arrive, here is the situation the brothers face:

  • What we know
    • Total camels: 35
    • Fraction to each brother: eldest: \frac{1}{2}, middle: \frac{1}{3}, youngest: \frac{1}{9}
  • What we want to find out
    • How many camels should each brother get?

In theory, this should be an easy problem: for the eldest brother, divide 35 by 2, and repeat for the others. Thus, the eldest brother should get 17 \frac{1}{2} camels — not too pleasant for the camel! And besides, half a camel is not that useful anyway. Clearly a better solution is needed.

Tahan and Samir arrive, Samir offers Tahan’s camel for the herd, and the problem changes. Now we have:

  • What we know
    • Total camels: 36
    • Fraction to each brother: eldest: \frac{1}{2}, middle: \frac{1}{3}, youngest: \frac{1}{9}
  • What we want to find out
    • How many camels should each brother get?

Now we’re getting somewhere.

36 divided by 2 is 18, 36 divided by 3 is 12, and 36 divided by 9 is 4. Thus, the three brothers get eighteen, twelve, and four camels, all of which give them full camels instead of useless fractional camels.*

Adding up all three brothers’ camelshare gives 18 + 12 + 4 = 34 camels, with two remaining from the herd. One was Tahan’s, one is now Samir’s. Everything is A-OK.

But where did that extra camel come from?

Re-read the father’s instructions again, carefully:

According to the express wishes of my father half of them belong to me, one-third to my brother Hamed, and one-ninth to Harim, the youngest.

At this stage, there are two ways to approach the problem. The slightly easier way is to convert the fractional shares. You can always multiply the top (numerator) of a fraction by any number, and the bottom (denominator) by the same number, and the fraction will be the same. One-half (\frac{1}{2}) is the same as two-fourths (\frac{2}{4}). So, let’s multiply each fractional camelshare by the number of camels, which is now 36. Thus, the father’s instructions now read:

According to the express wishes of my father \frac{18}{36} of them belong to me, \frac{12}{36} to my brother Hamed, and \frac{4}{36} to Harim, the youngest.

Or, if you prefer, you can convert the fractions to percentages (rounded to the nearest tenth of a percent):

According to the express wishes of my father 50% of them belong to me, 33.3% to my brother Hamed, and 11.1% to Harim, the youngest.

Either way, it quickly becomes clear: the father’s will was incomplete! The percentages don’t add up to 100%, so no matter how many camels were in the herd, some would be left over after the division.

Cool, huh?

*Useless Fractional Camels is the name of my They Might Be Giants cover band.

Except they weren’t: Malba Tahan

Camels_in_Ethiopia_01Except they weren’t: An occasional series about people who are Not What They Seem

Malba Tahan was a famous writer from Baghdad who traveled throughout the Middle East, recording tales of his adventures.

His most famous stories describe his travels with his friend Beremiz Samir, an Arabian traveler who was a mathematical genius. The pair traveled throughout the Muslim world like Watson and Holmes: Samir came up with ingenious solutions to practical mathematics problems, and Tahan recorded their adventures in beautiful, lyrical prose.

In 1949, soon after his death, Tahan’s work was published in Portuguese translation as O Homem Que Calculava (The Man Who Counted). It became an improbable bestseller in Brazil, where it remains one of its best-loved books. And so an unlikely hero to modern-day Brazilians is Malba Tahan, the Islamic Renaissance Man.

Except he wasn’t.Photo of Júlio César de Melo e Sousa, the man who created Malba Tahan

“Malba Tahan” was the fictional creation of Júlio César de Mello e Souza, a math teacher from Rio de Janeiro, who wrote the book to help teach his students how to solve word problems.

The result is beautiful, both in how Tahan/de Mello tells the tales and in how Samir/de Mello solves the problems. To appreciate the beauty, take a look at this, a translated version of one of the first stories in the book. It’s a bit long, but it’s definitely worth reading through:

We had been traveling for a few hours without stopping when there occurred an episode worth retelling, wherein my companion Beremiz put to use his talents as an esteemed cultivator of algebra.

Close to an old half abandoned inn, we saw three men arguing heatedly beside herd of camel. Amid the shouts and insults the men gestured wildly in fierce debate and we could hear their angry cries:

“It cannot be!”
“That is robbery!”
“But I do not agree!”

The intelligent Beremiz asked them why they were quarreling.

“We are brothers,” the oldest explained, “And we received thirty-five camels as our inheritance. According to the express wishes of my father half of them belong to me, one-third to my brother Hamed, and one-ninth to Harim, the youngest. Nevertheless we do not know how to make the division, and whatever one of us suggests the other two disputes.

Of the solutions tried so far, none have been acceptable. If half of 35 is 17.5, if neither one-third nor one-ninth of this amount is a precise-number, then how can we make the division?”

“Very simple,” said the Man Who Counted. “I promise to make the division fairly, but let me add to the inheritance of 35 camels this splendid beast that brought us here at such an opportune moment.”

At this point I intervened.

“But I cannot permit such madness. How are we going to continue on our journey if we are left without a camel?”

“Do not worry, my Baghdad friend,” Beremiz, said in a whisper. “I know exactly what I am doing. Give me your camel, and you will see what results.”

And such was the tone of confidence in his voice that, without the slightest hesitation, I gave over my beautiful Jamal, which was then added to the number that had to be divided between the three brothers.

“My friends,” he said, “I am going to make a fair and accurate division of the camels as you can see, now number 36.”

Turning to the eldest of the brothers, he spoke thus: “You would have half of 35 — that is 17.5. Now you will receive half of 36 — that is 18. You have nothing to complain about because you gain by this division.”

Turning to the second heir, he continued, “And you, Hamed, you would have received one-third of 35 — that is, 11 and some. Now you will receive one-third of 36 that is 12. You cannot protest as you too gain by this division.

Finally he spoke to the youngest, “And you young Harim Namir, according to your father’s last wishes you were to receive one-ninth of 35 or three camels and part of another. Nevertheless, I will give you one-ninth of 36, or 4. You have benefited substantially and should be grateful to me for it.”

And he concluded with the greatest confidence, “By this advantageous division, which has benefited everyone, 18 camels belong to the oldest, 12 to the next, and 4 to the youngest, which comes out to… 8 + 12 + 4 = 34 camels. Of the 36 camels, therefore, there are 2 extra. One, as you know, belongs to my friend from Baghdad. The other rightly belongs to me for having resolved the complicated problem of the inheritance to everyone’s satisfaction.”

“Stranger, you are a most intelligent man,” exclaimed the oldest of the three brothers, “and we accept your solution with the confidence that it was achieved with justice and equity.”

The clever Beremiz, the Man Who Counted, took possession of one of the finest
animals in the herd and, handing me the reins of my own animal, said, “Now, dear friend, you can continue the journey on your camel, comfortable and content. I have one of my own to carry me.”

And we traveled on towards Baghdad.

It’s a beautiful story, but how TF does the math work out? How does that make any sense?

Think about it, and then see the exciting conclusion!

P.S. I just wrote a post about math that ends on a cliffhanger!

P.P.S. And you just read that post. All the way to the end.

Image credits

Adorable camels from wikipedia user Bernard Gagnon
Photo of Júlio César de Melo e Sousa from Instituto Malba Tahan

An Unexpected Border

The world’s most unexpected border: these two countries seem to be a world apart, but high in the Hindu Kush mountains is a curvy 50-mile border between Afghanistan and China.

A screenshot of Google Earth showing the Afghanistan-China border

The only sane place to cross is the illegal unmarked border at Wakhjir Pass (15,780 feet), used only by the occasional drug smuggler:


This is also the largest time zone jump in the world (excluding the International Date Line) – cross into China at noon, and suddenly it’s 3:30 PM.

Russia doesn’t care

Boris, Natasha, and Fearless Leader from an episode of The Adventures of Rocky and Bullwinkle and FriendsAnother news day, another insight into the role of social media in the 2016 election cycle. I’ve written before about the role that fake social media accounts originating in Russia played in providing support for Donald Trump and other candidates. Unsurprisingly — at least in retrospect — those social media efforts went beyond support for specific candidates.

In a new study published today (Broniatowski et al., 2018) and reported on by many reliable media outlets, researchers at George Washington University studied three years’ worth of tweets containing keywords related to vaccines.

The reason that vaccines are important is that there is an ongoing debate in the U.S. about whether vaccines cause autism (they don’t), whether the initial study saying they did was a deliberate fraud (it was), and whether parents should vaccinate their children (they should). I have a lot of sympathy for parents who are reluctant about vaccinating their children — injecting your children with known disease-causing agents is undeniably creepy. But it works, to protect them and to protect other children too.

I haven’t read the full paper yet, but the bottom line is that during the period from 2014 to 2017 (and of course continuing through to today), Russian agents used (and continue to use) the vaccine debate to shift how Americans talk about social and political issues. But there is a fascinating difference between the candidate-based study I wrote about before and this one:

When discussing vaccines, the Russian trolls took both sides in the debate.

They didn’t care.

This at last provides some insight into why the Russians have invested time and energy into running social media campaigns in the U.S. They are seeking to divide our country, because a divided country is a weak country.

Don’t let them do it. Find someone who disagrees with you and talk to them, right now.

The Flag Test

A man holds the flag of Nepal at a paradeIf there’s one thing I am obsessed with — other than democratizing science, learning from data, evidence-based practice, people who aren’t what they seem, exploring the world with Google Earth, and Australian Rules Football — it’s flags.

A flag is a symbol of a group of people, a source of pride, and looks beautiful waving in the breeze. A flag represents all the best and worst impulses of humanity. When they all come together, like at the United Nations, you get a real sense of how we all come together as a world.

I’m not alone; there is a large and somewhat nerdy community of flag lovers called vexillophiles, from the Latin for “lover of flags.” We have an unofficial website, multiple YouTube channels, and an international organization (which of course has its own flag.

In my interactions with fellow vexillophiles, I have discovered an informal, tongue-in-cheek personality test, which is:

What do you think of the flag of Nepal? Is it kind of cool, or is it an Abomination Unto Flags?


The flag of Nepal, shown in the photo above and the illustration here, is the world’s only non-rectangular national flag (although there are some sub-national nonrectangular flags, most famously the state of Ohio and the city of Tampa, Florida). It consists of two red triangles, the bottom one slightly larger. The top triangle has a stylized, symbolic drawing of the moon, and the bottom has a similar drawing of the Sun. Both triangles are outlined with a thin blue border.

To be in the first category, you don’t have to like the flag, it’s enough to say, “that’s kind of cool, I respect what they were going for there.” Being in the second category requires strong opinions about what does or does not constitute a “real” flag.

Your opinion on the Nepal flag correlates with many other things, especially your opinion about whether people should get off your lawn. It’s not an absolute predictor, and of course correlation is not causation, but it’s an effect that I have noticed.

Zero guesses which side is the “get off my lawn” side, and zero guesses which side I’m on.

(Spoiler alert, highlight to reveal: It’s cool.)

Nepal flag photo: Hom Lamsal, Nepal Republic Media
Nepal flag illustration: Wikimedia users Pumbaa80 and Achim1999.

Except they weren’t: Iron Eyes Cody

Except they weren’t: An occasional series about people and things which are Not What They Seem

A middle-aged Iron Eyes Cody, dressed in a traditional Native American cloak and with a feather on his head, in an undated publicity photoTo three generations of movie fans, Iron Eyes Cody was THE Hollywood Indian. He was born in Oklahoma in 1904 to a Cherokee father and a Cree mother. He spent his youth performing in traveling Wild West shows, where he taught himself the sign languages of other Nations. In 1924, he moved to California, and within two years was appearing as an uncredited extra in Hollywood.

His career took off from there, and he eventually appeared in more than 200 films and TV series, particularly Westerns. He played in films with A-list actors like John Wayne (The Big Trail in 1930) and Steve McQueen (A Man Called Horse in 1970). But his most famous role came at age 65 in a Public Service Announcement TV commercial that was an early advocate for environmental conservation movement. It’s horribly dated now, but it had a real impact on changing public attitudes:

Cody wrote an autobiography, died in 1999 at age 94, and is buried in the “cemetery of the stars,” Hollywood Forever Cemetery. He is in the mausoleum with his beloved wife Bertha, not far from stars like Victor Fleming, James Garner, and Marilyn Monroe.

Over a career spanning nearly 70 years, Iron Eyes Cody’s career perfectly traced America’s changing attitudes toward the people known first as Indians, then as American Indians, then as Native Americans — all the while staying true to his heritage as a Native American.

Except he wasn’t.

He was born as Espera Oscar de Corti in small-town Louisiana, the son of two immigrants from Sicily who ran the town grocery store. He moved to California at 19, where he used his dark skin, talent for telling a good story, and genuine acting talent to score a long and successful career as an actor.

The truth began to come out in 1996, when his half-sister gave an interview to the New Orleans Times-Picayune newspaper. de Corti/Cody denied the rumor, but it was officially confirmed after his death.

What do we make of his story? Was this the worst kind of cultural appropriation, the story of a white man literally taking on a fake Native American identity? Was it a well-meaning fib that had a happy ending and actually did some good? Did it start out for convenience, but then eventually de Corti managed to convince himself he really was Cody?

If it helps: he married a for-reals Native American woman, adopted two children from reservations in his fake-home state of Oklahoma, and spent much of his life advocating and fundraising for Native-led charities and causes.

Questions like these are why I find these except-they-weren’t stories so fascinating.

What do you think?