Three Days in Dwarfland (and sequels)

In Soviet Russia, math is looking forward to you!

Vladimir Arthurovich Manasevich was born in 1904 in Baku, the capital of Azerbaijan, as a son of a millionaire and hereditary honorary citizen[52], Arthur Borisovich Manasevich. In 1906 the Manasevich family moved to Moscow. The whole family was musically gifted; Vladimir recalls that at their home, someone was playing an instrument or singing almost all the time. Vladimir received music lessons, and planned to become a composer. He also studied French[53], and physics.

Then came the October Revolution. Everyone was required to find a worker’s job; even Vladimir needed to get a job first, so that he would be allowed to study at a university later. His father became a salesman in a cooperative store, and his mother worked in a nursery. Vladimir tried many things; he cooked soap, made combs, filled cartridges at a tobacco factory, and worked as a folder in a printing house. During that time, he also studied sculptures and painting, and at the age of sixteen he became a studio member of the Moscow Chamber Theater.

As an eighteen year old, Vladimir legally changed his surname to Levshin, and switched to science. He graduated from Mendeleev's Institute of Chemical Technology. For over 40 years he taught higher mathematics, strength of materials, and theory of elasticity at Moscow universities.

But even as a respected professor and scientist, Levshin continued doing art in his free time. He wrote one-act plays for pop artists and funny skits for circus clowns. In 1953 he wrote a fairy tale for children, “The Adventures of the Braggart Cat”. Then, as his wife describes it:

He once wrote a children's fairy tale about the Braggart Cat. The story was heard on the radio. Then they recorded it on a record. And then Vladimir Arturovich met the writer Lvovsky[54] who told him: “You are a mathematician. You write for children. Why don’t you write to children about mathematics?”

Following the advice, in 1962, Levshin wrote his first book for children about mathematics: “Three Days in Dwarfland”. The book was reprinted several times, and a radio play was based on it. It was followed by two sequels, another trilogy, and a few more books.

Many of the books were co-authored by his wife, Emilia Borisovna Alexandrova. She described her role like this:

You may ask: why does a mathematical writer need a non-mathematician co-author? I will also answer this with a question: why does he need a mathematician co-author? After all, he is already an expert in his field! Everything is clear to him!

But the rub is that it is sometimes difficult for an expert to imagine the train of thought of a non-expert. He does not always foresee what exactly can baffle the know-nothing. And when you write for children, it is necessary to anticipate this.

Vladimir Arthurovich Levshin died in 1984 in Moscow. Emilia Borisovna Alexandrova survived him by ten years and died in 1994.[55]

Technically, there actually was a section on mathematics already in the Braggart Cat. The protagonist meets three bunnies, and…

😼 Hey bunnies, what are you doing here?

🐰 Sharing carrots. 🐰 It does not work. 🐰 One is always missing.

😼 That’s nothing, you just need to know arithmetics.

🐰 And you do know how to divide?

😼 I know everything. I have passed seven classes, didn’t miss a single one.[56]

🐰 Help us! It does not work.

😼 Okay. What do you have, five carrots?

🐰 Five.

😼 And how many are you? Count…

🐰 One, 🐰 two, 🐰 three.

😼 So, it is clear. Only five carrots; two times two is four, one is extra; needs to be thrown out into the river 🥕, like this.

🐰 We are not four, but three.

😼 That means there are two more extra, we will also throw them out; one 🥕, two 🥕, like this, and now we will divide. Hold it, one; hold it, the other.

🐰 And me?

😼 What?

🐰 What about my carrot?

😼 Again not enough? Oh, you have confused me completely. I have no time for you, I’m going away…

🐰 (sadly) So much for arithmetics.

Let’s just say that the protagonists of the later books got more competent  at math.

Three days in Dwarfland (Три дня в Карликании) is the first book in a trilogy. It starts with an unspecified narrator (Levshin himself) inviting three children - Tanya, Seva, and Oleg - to Dwarfland, a mysterious country, where the inhabitants drink ink and wash themselves using erasers.

The children find themselves in a land of anthropomorphic digits (and other beings), an “arithmetic state”. Walking around, they see various interesting things, and often find a willing guide who explains.

Mathematical explanations are mixed with silly stories, creating a unique experience. We meet colorful characters, such as little Zero, a mischievous boy who often gets in trouble, but is enthusiastic about learning (and in later books becomes one of the protagonists); his mother, a fat Eight; a young teacher Four with a ribbon in her hair; a sea captain One; etc.

This book is not a textbook; the author is not trying to teach us math systematically the way they do it at school. He just keeps mentioning interesting things related to math, and places them into the context of a story (which is mostly just: kids walking around, being curious). Sometimes he actually explains, e.g. that exponentiation means repeatedly multiplying the number by itself, and calculating a root is the opposite of that. Sometimes he just hints at more complex topics, e.g. that if we keep dividing something by smaller and smaller numbers, the results get bigger and bigger, and therefore a division by zero would give us infinity (but this is described as a dangerous thing that the citizens of Dwarfland usually try to avoid). Sometimes he paints a vivid visualization, e.g. a division of fractions is illustrated by acrobats jumping on a trapeze (the trapeze representing the fraction line). And sometimes he just says something silly, e.g. the origins of arithmetic are “explained” by prehistoric people needing to put the meat in a warehouse after the hunt.[57]

Sometimes the reader is invited to solve a problem along:

        “Oh, what beautiful apples are in this garden! Could we taste them?”

“Why not,” said Four, “but to do that, the apples must first be picked.”

“We wanted to, but we couldn’t. Too high.”

“Perhaps you don’t know our rule? The apples will fall to the ground on their own - you just have to solve some problem.”

To our surprise, the kids were not at all upset. They busily took out small sticks from their pockets and prepared to write down the conditions of the problem in the sand.

“So,” continued Four, “there are apples on three plates. The first plate contains half of all the apples. When half of what was on the second plate was taken from the first one, and then half of what was on the third plate, only two apples remained on the first plate. The question is, how many apples were on each plate at the beginning? Is it clear?”

The kids began to snort in concentration, moving their sticks on the sand, some even stuck out their tongues in zeal. Soon, however, their mood clearly deteriorated. Many even started crying. The Four was not at all surprised by this, took out a dazzlingly white handkerchief and carefully wiped their small wet noses.

“There is no need to cry,” she said, “this task is still difficult for you. Let our guests solve it. And then we will all taste the wonderful apples together.”

“Tanya, all hope is in you!” Seva whispered. He had long been impatient to get to know the apples better.

Very soon the garden was literally filled with ripe fruits.

(...)

“Absolutely correct!” said Four, looking at the numbers, and immediately erased them with her foot.

“Why, why did you do that?” the kids squealed.

“Let the one who has not solved this problem make sure he solves it himself.”

Topics mentioned in the book:

• mathematical operators (“periods, commas, small dashes, large dashes, crosses, round brackets, square brackets, curly brackets and many, many more completely incomprehensible signs”)
• exponentiation and roots
• the origin of Arabic numerals in India
• infinity (“something that has no end … no matter how far you walk, you are always in the very middle”)
• binary search
• commutativity of addition
• multiplication by zero, infinity times zero
• divisibility rules
• prime numbers
• perfect numbers, amicable numbers
• Roman numerals, hieroglyphs, Cyrillic numerals[58], Babylonian cuneiform numerals
• comparing fractions
• division by zero
• dividing fractions
• positive and negative powers of 10
• repeating decimals
• famous historical mathematicians[59]
• usefulness of math at discovering planets, etc.

Target audience… I would guess, children between 8 and 12 years old? Not sure.

The Black Mask from Al-Gebra (Чёрная Маска из Аль-Джебры) is the second book in the Dwarfland trilogy, and also the first book written by Vladimir Levshin in cooperation with Emilia Alexandrova.

The protagonists of the previous book - Tanya, Seva, and Oleg - receive a telegram from little Zero - the mischievous boy from Dwarfland - asking them for help. The children visit Dwarfland again, this time also taking their dog, Donut.

Little Zero describes his meeting with a mysterious masked stranger who claims to be cursed. The stranger does not remember his identity, and cannot remove the mask, until his secret is revealed. He gives little Zero a magical pea pod which gradually provides hints.

The magical pod reveals a long secret underground passage leading somewhere out of Dwarfland. Little Zero is too small to join such an adventure, and he regretfully stays at home. The children promise to keep sending him letters, delivered by Donut. The rest of the book consists of their mutual correspondence. The children experience various things in the neighboring state of Al-Gebra, which is similar to Dwarfland, but inhabited by anthropomorphic digits and letters. Meanwhile, little Zero tries various things at home (often recruiting other little zeroes for help) inspired by the children’s messages.

While this book is similar in style to the previous one, there are also some noticeable differences. In the first book, the common theme was “arithmetic” (addition, subtraction, multiplication, and division), in the second one it is “algebra” (using letters in place of some numbers). In the first book, things were either explained or demonstrated by analogies. In the second book, there are also enthusiastic proposals made by little Zero… some of which turn out to be good ideas, while others turn out to be wrong.

Towards the end of the book, the children master the techniques of al-gebr and al-muqabala[60] and rescue the cursed stranger.

Topics mentioned in the book:

• substitution cipher

• arithmetic and geometric means

• number line, arithmetic operations using negative numbers

• Latin[61] and Greek letters

• decimal numbers, irrational numbers, complex numbers

• zero power, zero to the power of zero, half power, negative power

• using letters to represent numbers

• factorials

• polynomials

• multiplication of powers

• Pascal’s triangle

• arithmetic progression, the sum of arithmetic progression

• geometric progression

• how to solve a linear equation

• powers of negative numbers

• powers of imaginary unit

Captain One’s Frigate (Фрегат капитана Единицы), in later editions called Little Zero the Sailor (Нулик — мореход) is the final book in the Dwarfland trilogy. The three children from the previous books do not occur here; the stage belongs to little Zero and his new friend Pi.

Little Zero decides to finally get an adventure of his own, and joins the crew of captain One, sailing the arithmetic, algebraic and geometric seas and oceans. The rest of the story is similar to the first book: the ship sails, random things happen; only now little Zero is the curious traveler, and most of the things observed on various islands are geometric in nature.

Topics mentioned in the book:

• axioms (if you don’t take them on faith, you can make the gods really angry)
• lines, line segments
• angles: right, acute, obtuse; angular degrees
• Pythagorean theorem
• congruent triangles, similar triangles
• proper and improper fractions, decimals
• probability, statistics
• units of measurements
• volumes of solids
• integral (the concept of the area under a curve)
• circle: radius, diameter, sector
• Möbius strip
• use of math in music
• percents
• altitudes and orthocenter, angle bisectors and incenter, medians and centroid
• functions
• cycloid
• coordinates
• ellipse

And thus concludes the Dwarfland trilogy.

Wait, there is more!

Levshin wrote a few more books, and I originally assumed that those were written in “universes” other than Dwarfland. Which was mostly correct… except for another trilogy. But now I am running out of time, so I will keep this section brief.

The Dwarfland trilogy is followed by the Absent-Minded Master trilogy, consisting of:

• Thesis of the Absent-Minded Master (Диссертация рассеянного магистра)
• Travel Notes of the Absent-Minded Master (Путевые заметки рассеянного магистра)
• In Search of the Stolen Stamp (В поисках похищенной марки)

The protagonist of this trilogy is an extremely absent-minded mathematician trying to write his master thesis. After introducing himself to Levshin and asking him to take care of his pet dog (which turns out to be a cat), he departs to Dwarfland. One year later, he sends the completed thesis for a review. Levshin does the smart thing and outsources the review to children - Tanya, Seva, Oleg, and little Zero.

Each chapter in the trilogy consists of a chapter from the “thesis”, which is a description of the protagonist’s adventures, containing lots of mathematical mistakes, followed by a discussion of children reviewing the chapter.

Trying to find the right words that would explain my fascination with Levshin’s books, I remembered a quote by Antoine de Saint-Exupéry:

“If you want to build a ship, don’t drum up the men to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea.”

I believe this is exactly what Levshin has achieved with his Dwarfland trilogy.[62] Although the books may teach you a few new math concepts, the most important part is their “hidden curriculum” which says: math is a vast and exciting place full of interesting things to discover.

For those who speak Russian, here is some audio on YouTube:

• Braggart Cat
• The Adventures of little Zero: Part 1; Part 2; Part 3