The library
of Rajawali High was not merely a place to store books. It was a sanctuary—a
holy site for those who worshipped academic grades above all else. The floors
were lined with thick sound-dampening carpets, the shelves were made of
polished mahogany, and the air conditioner was set to a temperature low enough
to freeze the intentions of any student thinking of skipping class.
In a corner
of the room dubbed the "Competition Zone," fifteen selected students sat around
a large round table. This was a special tutoring session for the city-level
National Science Olympiad preparation.
The
atmosphere at the table was tense. The only sounds were the scratching of pens
on paper and the occasional sigh of frustration.
At the head
of the table, Nadia sat upright like a queen presiding over a cabinet meeting.
Spread before her were three thick English reference books, a high-end
scientific calculator, and a pencil case organized by color gradient. Her eyes
stared sharply at the problem sheet in front of her, as if she wanted to drill
a hole through the paper with laser vision.
Opposite
her, a painful contrast was on display. Salim slouched in his chair, his legs
stretched out under the table (luckily not kicking anyone). There were no
reference books. Only a standard pen with its cap missing, and a piece of
crumpled scratch paper he had used to make a paper airplane earlier.
Mrs. Laras,
the young yet perfectionist Olympiad team coordinator, tapped on the small
whiteboard beside the table.
"Alright,
pay attention," Mrs. Laras's voice broke the silence. "Last problem for today.
This is a combinatorics problem that appeared in the national team selection
back in 2018. I want to see who has the sharpest logic here."
Mrs. Laras
wrote down the problem. Not numbers, but a dizzying logic puzzle.
"There are
20 people in a room. Each person has exactly 3 enemies in that room. Prove that
we can divide these 20 people into 2 groups, such that every person has at most
1 enemy in their own group."
Nadia
immediately grabbed her pen. Her eyes sparkled. This was her favorite type of
problem: Graph Theory. She had memorized the formulas; she knew the theorems.
Meanwhile,
Salim yawned. He spun his pen between his fingers—a pen-spinning trick he
learned from YouTube while bored waiting in line for cheap groceries.
"What's
wrong, Lim? Giving up?" a cynical voice came from beside Nadia. It was Rinto.
He was there not because he was a math genius, but because his parents were the
library's biggest donors, earning him an "honorary seat" on the Physics
Olympiad team, which happened to practice in the same room.
Salim turned
his head lazily. "Nah, Rin. I'm just thinking, if everyone has 3 enemies, these
people must have a really toxic social circle."
A few
students suppressed a laugh, but Nadia slammed her hand lightly on the table.
"Can you be
serious?" Nadia hissed. "This is a selection, Salim. If you just want to play
around, you better leave. Give that seat to an 11th grader who actually cares."
Salim
shrugged. "I am serious. Seriously waiting for you to solve it."
Nadia
scoffed. She refocused. She began drawing dots (vertices) representing people,
and lines (edges) representing enmity. She tried to use the Pigeonhole
Principle combined with Turan's Theorem.
Nadia's
paper was filled with complex graph scribbles. She tried to prove it using
mathematical induction.
Let P(n) be
the statement...
If we divide
set V into V1 and V2...
Fifteen
minutes passed. Sweat began to bead on Nadia's forehead. Her proof was stuck
halfway. Every time she moved one "person" to the other group to reduce
enemies, the number of enemies for someone else increased. It was like
squeezing a water balloon; press one side, it bulges on the other.
"Two minutes
left," Mrs. Laras's warning made the atmosphere even more gripping.
Nadia
panicked. She erased part of her answer, trying a greedy algorithm approach.
But there wasn't enough time. Her logic was tangling up on itself.
"Done," a
flat voice spoke out.
It wasn't
from Nadia. It was from across the table.
All eyes
turned to Salim. The scratch paper in front of him was still completely blank.
Clean. No formula scribbles, no graph drawings.
"Where is
the answer, Salim?" Mrs. Laras asked, raising an eyebrow. "Your paper is
empty."
"The answer
is in my head, Ma'am. Writing it down makes my hand tired," Salim answered
casually.
Nadia
laughed dismissively. A forced laugh. "Don't be ridiculous. Proof problems must
have written steps. You can't just guess 'Proven' or 'Not Proven'. This isn't
multiple choice."
"Who said I
was guessing?" Salim straightened his posture. His "joker" face vanished,
replaced by a sharp, cold gaze—a side of him rarely shown unless his ego was
provoked.
"Okay, Queen
of Formulas," Salim said, looking straight into Nadia's eyes. "You tried using
induction, right? And then you got stuck because you were confused about
managing the overlapping enemy variables?"
Nadia fell
silent. Salim's guess was dead on.
"You're
overthinking it," Salim continued. He took his pen, then drew a large circle on
his scratch paper, dividing it into two.
"Mrs. Laras,
may I explain verbally?"
"Go ahead,"
Mrs. Laras crossed her arms, intrigued.
Salim
pointed to the two sections of the circle. "Let's just assume we divide those
20 people randomly into two groups first. Group A and Group B. Just random,
free choice."
"That
doesn't prove anything, idiot," Rinto interrupted.
"Quiet for a
sec, Donor," Salim cut in sharply, making Rinto's face turn beet red. "Listen."
Salim
continued, "Now, let's check each person. Let's say there's a guy named Budi.
If Budi has, say, 2 or 3 enemies in his own group (let's say Group A), what
should we do?"
Salim looked
at Nadia, fishing for an answer.
"Well...
move him?" Nadia answered hesitantly.
"Exactly,"
Salim snapped his fingers. "Move Budi to Group B. Since he has a total of only
3 enemies, if he has at least 2 enemies in Group A, that means in Group B he
has at most 1 enemy. So, when he moves, his condition becomes safe (max 1
enemy)."
"But..."
Nadia interjected, her brain spinning fast to find a loophole. "If Budi moves
to B, he might increase the number of enemies for someone already in B, right?
That would ruin the stability of others in B."
"And that's
the point," Salim smiled crookedly. A smile that made Nadia feel small. "That's
the key. We don't need to care about the stability of others yet. We use the
variable: Total Number of Enemy Pairs in a group."
Salim wrote
a simple symbol: E_total.
"Every time
we move a person who has 2 or more enemies in their group to the other side,
the total number of enemy pairs in that group must decrease. Why? Because we
sever 2 enemy ties in the original group, and only add a maximum of 1 enemy tie
in the new group. Minus 2 plus 1 equals minus 1. The total 'conflict' in the
room decreases."
Salim put
down his pen. "Since the number of enemies is finite (not infinite), this
reduction process cannot go on forever. Eventually, this process will stop.
When does it stop? When there is no one left who has 2 or more enemies in their
group. At that moment, the problem's condition is met. Proven."
Silence.
A long,
painful silence for Nadia.
Salim's
explanation was so simple, so elegant, and so... human. He didn't use complex
sigma notations or theorems from thick books. He used movement logic that even
a middle schooler could understand.
Mrs. Laras
smiled widely. "Descent Algorithm. Or proof by Invariant. Absolutely brilliant,
Salim. You solved a complex problem by shifting the perspective into a simple
optimization problem."
Nadia stared
at her paper filled with useless scribbles. Her hands trembled, holding back
anger and shame. Again. She lost again. She studied at the most expensive cram
school in the capital every weekend, she devoured imported books, but this
scholarship kid whose bike often broke down was always one step ahead of her.
"That...
that's cheating," Nadia muttered softly, her voice shaking.
"What's
cheating, Nad?" Salim asked, his tone back to being casual and slightly
mocking. "Logic can't be cheated. You were just too busy memorizing the map
that you forgot to look at the road."
Nadia stood
up abruptly, her chair screeching loudly against the floor. She packed her
books with quick, rough movements. "I have a headache, Ma'am. Permission to
leave early."
Without
waiting for Mrs. Laras's answer, Nadia stomped out of the library. Her
footsteps were heavy, voicing the defeat of her ego. Rinto hurriedly stood up,
giving Salim a murderous glare, then ran to chase after Nadia. "Nad! Wait! Just
ignore that poor guy!"
Salim just
shook his head watching the drama. He slouched back into his chair.
"You were
too hard on her, Lim," Mrs. Laras commented, though her tone wasn't really
scolding.
"I just
answered the question, Ma'am. If she takes my answer as a personal attack,
that's between her and her ego," Salim replied indifferently.
Suddenly,
Salim's phone in his pocket vibrated. A message came in. From Dani.
Danifridge:
Oy, Lim! Hurry to the parking lot. Your bike got peed on by a stray cat. No
wait, your bike chain came loose again. I'll wait for you, just hitch a ride in
my car. Rizki and Maya are coming too.
Salim smiled
thinly. The contrast of his life was indeed funny. Inside this room, he was an
intellectual giant who had just humiliated the school's "queen." But once he
walked out that door, he was just Salim, the scholarship kid with a broken bike
who had to hitch a ride in his friend's luxury car.
He got up,
grabbing his blank scratch paper—which to others was trash, but to Salim was
proof that he didn't need tools to think.
"I'm heading
out first, Ma'am. Diplomatic matters. Motorcycle chain matters," Salim excused
himself.
As Salim
walked down the increasingly empty school hallway, he didn't realize that
behind a corridor pillar, Nadia was standing there. Her eyes were red from
holding back tears, her hand clenching her math book tightly.
The hatred
in Nadia's eyes was no longer just academic rivalry. It was a grudge. A grudge
from feeling her dignity stripped away layer by layer by someone she considered
"beneath her."
"Just you
wait, Salim," Nadia whispered to the empty air. "Someday, there will be a test
you can't solve with your relaxed logic. And when that time comes, I'll be the
one laughing the loudest."
Unbeknownst
to Nadia, her prayer would be answered sooner and more horribly than she
imagined. The test wouldn't be on paper, but on life itself. And in that test,
Salim's relaxed logic might be the only thing separating them from a grave.
Salim
reached the parking lot. Dani was already waving from inside his gaudy yellow
Mustang. Rizki sat in the front calmly, while Maya sat in the back, leaving an
empty space beside her for Salim.
"Took you
long enough, Professor!" Dani shouted. "Did you just crack another atomic
formula?"
"Just the
formula for raising Nadia's blood pressure," Salim answered as he got into the
car, welcomed by Dani's laughter and Maya's gentle smile.
The car door
closed, isolating them in air-conditioned luxury, leaving Rajawali High
standing grandly with all its intrigues and ambitions behind. For a moment,
life felt normal. Warm. Safe.
But it was
only the calm before the storm. The clock kept ticking closer to the Study Tour
departure day. The day when all formulas, all rivalries, and all social
statuses would be reset to zero.