Translator: Dreamscribe
Monday,
Yu Seo-ha woke up earlier than usual.
The dawn air seeping through the window was cold. Mid-November, winter was already right around the corner.
Today was the first preliminary round of the Korean Mathematical Olympiad.
Seo-ha sat up from his blanket and opened the window. The cool air seeped deep into his lungs, completely waking him up.
He changed into light workout clothes and quietly stepped outside.
At 6 a.m., the village was quiet, but here and there he could sense the presence of people preparing for the day. Seo-ha began to jog with light strides.
There was a reason why Seo-ha had started running.
The mathematician Ramanujan,
Died young at the age of 32 from malnutrition and tuberculosis.
The cause was an extremely irregular lifestyle and lack of exercise. He sat for long hours every day, immersed only in mathematical research.
Mathematical history was an incredibly interesting storybook to Seo-ha.
Following the development of Eastern and Western mathematics and peeking into the lives of great mathematicians was a joy above all.
How Gauss, at the age of seven, instantly calculated the sum from 1 to 100, the story of how Euler continued his research even after losing his eyesight, and even the absurd tale of Galois dedicating himself to organizing his mathematical theories on the eve of a duel.
It was fascinating to Seo-ha that they, too, had once been children like him, had moments when they fell in love with the charm of mathematics, and sometimes felt despair and sometimes ecstasy.
Seo-ha occasionally even felt as if he could transcend time and space to converse and breathe with them.
‘If only Ramanujan had lived a little longer...’
The loss his death brought to the mathematical world was incalculable.
Among the approximately 3,900 formulas and theorems he left behind in his short life of 32 years, many were not proven to be true until 60 years after his death, at the end of the 20th century.
It was only after the development of computers that the precision of his intuition was revealed.
‘What if he had lived to be 70?’
Despite his short life, the theories he researched, such as infinite series and continued fractions, became the foundation of modern cryptography and computer science. And the theta function he mentioned in his last letter is now being used to determine important dimensions in string theory and M-theory, the biggest topics in modern physics.
What’s most unbearable is that there are still many aspects of his theta function that remain undiscovered.
‘What a terrible personality.’
Why do mathematicians just toss out such important problems without even proving them?
Seo-ha couldn’t understand them.
How could they so confidently release unfinished ideas to the world?
Ramanujan, in particular, often omitted the proof and only wrote down the results, claiming that the goddess Namagiri told him the formulas in his dreams.
Yet there were almost no errors, which drove Seo-ha crazy.
The history of mathematicians he had read was, in short, a fight against health.
So he runs.
Huff, huff.
His breath was rising to his throat.
Once he actually started, he found that running had many benefits beyond just health.
His mind, which usually kept automatically calculating steps, eventually stopped. And the ideas that naturally surfaced while running.
There were times when he found answers during a run that he couldn’t see while sitting and agonizing.
It had been half a year since he started running.
At first, he couldn’t even run for ten minutes without gasping, but now he could easily go for thirty minutes.
‘Is this what endorphins feel like?’
He saw someone in the distance.
“Seo-ha! Take some cucumbers and tomatoes with you!”
Mr. Choi, who ran a greenhouse, waved at Seo-ha.
“Thank you, sir! My mom will love them!”
Seo-ha was already a well-known figure in Okcheon.
Seo-ha, having run around the village, returned home. His whole body was properly warmed up and his mind felt neatly organized.
“Welcome back! What did they give you again? Go wash up and let’s have breakfast!”
Mi-young took the bag from Seo-ha’s hand.
After a shower, he sat at the table to find a more plentiful breakfast than usual prepared.
‘Looks like my parents are more nervous than I am.’
“Mom, you didn’t have to go out of your way... just our usual meal would’ve been fine...”
“Still, today’s an important day. You should eat properly before going.”
In fact, Mi-young hadn’t been able to sleep properly the night before.
She knew well that her son was exceptional, but still, such is a parent's heart. Half-asleep and preparing breakfast, she ended up making more side dishes than usual without realizing it.
“Wow! There’s so much delicious food!”
Still half-asleep from waking up early, Seo-eun screamed and sat down at the table.
Looking at Seo-eun, Chul-ho said,
“Mom prepared all this so oppa can do well on his exam.”
“Then there’s nothing to worry about. Oppa doesn’t have anything he doesn’t know. He always answers whatever I ask. Even Mom and the teacher often say they don’t know!”
At Seo-eun’s words, the whole family burst into laughter.
***
A high school in downtown Daejeon.
By the time Seo-ha arrived at the test site, many students were already there.
‘Wow! There are so many people.’
This year, the total number of participants in the domestic KMO was about 8,000.
Test sites were set up in 16 cities and provinces nationwide, and even in just the Daejeon area, where Seo-ha was assigned to, more than 500 students had gathered.
One school wasn’t enough, so they split the test among three high school buildings. It seemed like every student good at math in the country was participating.
Only 200 students would be selected from the first round, and making it into that group would qualify as a strong credential for applying to special admissions programs for talented students or science tracks in college admissions.
All around the hallways and classrooms, many students were already immersed in last-minute preparations.
A student seated in the front row had his hands clasped and eyes closed.
Another student was mumbling to himself, repeatedly reciting formulas written in his notebook.
It was a kind of tension Seo-ha had never seen before.
The proctor entered.
“We will now begin the first preliminary round of the Korean Mathematical Olympiad. The test time is three hours. There are 25 questions in total, all five-choice multiple-choice questions.
As this is an important exam to select talents to represent Korea, I hope everyone does their best without regrets.”
Beeeep-
The question papers and answer sheets were distributed, and the exam began.
‘Will there be new types of questions?’
It was said that the Olympiad’s first round often featured clever problems to identify truly outstanding students.
Seo-ha’s heart pounded.
Question 1.
[When four coins are tossed simultaneously, what is the probability of getting exactly two heads?]
A probability question.
It could be easily solved using combinations or binomial expansion.
Seo-ha skipped writing out the solution and went straight to selecting the answer.
Question 2.
[When coloring the four corners of a square using red, blue, and green, how many distinct coloring patterns are there?]
‘Haha!’
A coloring problem made him laugh.
He was reminded of the Four Color Theorem, which still hadn’t been fully resolved. Seo-ha hadn’t forgotten the problems that remained unproven.
Whenever a clue came to mind, he tested new possible approaches.
‘Compared to the Four Color Theorem, this is nothing.’
By applying Burnside's Lemma, the problem could be solved easily. The order of the rotation group is 4; calculating the coloring methods fixed by each rotation...
Seo-ha’s hand swiftly moved through the answers without hesitation.
As he progressed through the problems, the difficulty steadily increased.
Question 7 was cleverly designed to make it easy to overlook certain conditions, but it was no use against Seo-ha. That was because he perceived numbers not as fragments but as groups of patterns. Even if the conditions changed, the structure of the group as a whole remained unchanged.
Question 10 was the same. The problem was structured in a way that trapped you unless both solutions of the function were calculated. But Seo-ha recognized the roots not as numerical values, but as trajectories on a graph.
From question 11 onwards, the high-difficulty problems began in earnest.
A geometry problem using complex numbers, an analysis problem dealing with the limits of a sequence, graph theory, and so on.
‘Huh?’
The tip of Seo-ha’s pen stopped while solving a problem.
‘Something’s strange.’
1, 4, 2, 5, 3, 2, 5, 3, 1, 4, 3, 1...
Seo-ha stared blankly at his half-marked answer sheet.
The numbers from 1 to 5 were evenly distributed.
A 5 x 5 square grid was drawn in Seo-ha’s mind. Then, coordinates were assigned to the marked answers, and they began to rearrange within it.
In that moment, Seo-ha realized that the numbers arranged horizontally and vertically were oddly balanced.
‘Orthogonal array?’
It wasn’t a coincidence.
The probability of such numbers appearing by chance was 1 in 500,000. When considering the orthogonal pattern, it was less than 1 in 10 billion.
It was clear someone had intentionally embedded all the answers within a single combination structure.
Seo-ha pushed aside the test questions and began solving the puzzle.
A chessboard-like grid began filling with numbers in his mind. Going down each vertical line, the numbers did not repeat. Even across each horizontal line, the five numbers were evenly and subtly distributed.
‘Latin square.’
Seo-ha immediately formed a hypothesis.
By interpreting each question as a coordinate (i, j) on the grid and finding what values the coefficients a and b take, all the answers could be reconstructed.
‘The answer is (a.i + b.j) mod 5 + 1.’
Seo-ha set down his pen.
All 25 cells were filled with numbers.
It was a beautifully ordered structure, showing symmetry in every row and every column.
Seo-ha felt he already knew the answer to the next question. The position corresponding to coordinate (3, 3).
‘Number 4.’
He tested the hypothesis.
His hand sped up as he continued solving the problems.
The answer was, without fail, number 4.
A smile spread across Seo-ha’s face.
It was as if the invisible architect of the exam was speaking to him.
‘You see the pattern, don’t you?’
Each year, fewer than ten students get a perfect score on the first round of the KMO.
This was a message of praise from the problem setter, meant only for the students who answered every question correctly. Of course, the setter never expected any student to actually notice it.
But Seo-ha had completed the entire grid in the middle of the exam.
Questions 14, 15, 16...
There was no need to solve them. The answers were already in front of him.
He entered the answers without going through calculations. And every time he checked, they were correct.
Up to question 24, everything matched without a single error.
And then came the final, question 25.
‘Number 5?’
The answer, which should clearly have been number 1, was pointing to a different number.
Seo-ha’s right eyelid twitched.
There was no way the grid formula could be wrong. It had worked perfectly for 24 questions, why only question 25?
‘Did I miss something?’
He retraced the entire process in his mind. But there was no error anywhere.
Just in case, he gripped his pen tightly and re-solved question 25 from the beginning.
Polyomino.
A combinatorics problem using rotation, reflection, and symmetry.
But still, the answer was number 5.
Thump, thump.
Seo-ha’s breath gradually became rough.
‘This isn’t a sequence or a puzzle. It’s an exam where I have to write the answer.’
He knew that in his head, but his hand trembled as he tried to mark the answer.
‘Just mark number 5 and finish this already.’
He shut his eyes tightly and brought the marker to the answer sheet.
But he couldn’t fill it in.
An imperfect pattern, an unexplained exception.
He couldn’t accept that and ruin the arrangement.
Cold sweat trickled down Seo-ha’s forehead.