Chapter 30: Critical Points and Mediums

Raindrops slammed against the glass. At first they were scattered. Very quickly they joined into lines. Trails of water warped the cinder track outside the window. The air pressure in the classroom seemed to sink. The sound of pages turning grew heavier. The scratching of pen tips across paper grew denser. The news that scholarship slots had been cut was like a stone. Thrown into the water. Before the ripples had even spread. It had already sunk to the bottom.

Lin Chen stared at the “20%” on the blackboard. His pencil drew three horizontal lines across his scratch paper. 88 points. Not a slogan. Arithmetic. Sixty points from the basic problems. They had to be all correct. Twenty points from the medium problems. He needed eighteen. Twenty points from the final major problem. He needed ten in method marks. Total score: 88. Margin for error: zero.

He lowered his head. Looked at page twelve of the exercise book. His fingers pressed down on the paper. To keep the dampness seeping in from the rain from wrinkling it. He began breaking time apart in his head. Twenty minutes of morning reading. Ten minutes between classes. Forty minutes at noon break. Two hours of evening study hall. Forty-five minutes after lights-out. Put the fragments together. Four and a half hours. Four and a half hours. Enough to digest the first three chapters of mechanics. Trigonometric functions. Oxidation-reduction. Classical Chinese content words. He picked up the eraser. Rubbed out the guide lines on the scratch paper. Drew them again. Force analysis. Orthogonal decomposition. Set up equations. Substitute. Check the arithmetic. No step could be skipped. Skip one step. Lose points. Skip two. Lose the problem. He did not know the county high school’s grading standard. But he knew this. Fill in every step. If the grader could not find where to deduct points. That was points.

The physics teacher began explaining forces on an inclined plane. The chalk sketched a rough set of axes on the board. Chalk dust fell onto his sleeve. Lin Chen did not look up. He stared at his own notebook. Broke the teacher’s board work into three columns. Known conditions. Hidden conditions. Goal of the solution. He underlined with different colored pencils. Blue for what was given. Red for what was implied. Black for the target. Colors to separate. Logic in layers. Not relying on memory. Relying on structure.

A boy in the front row turned around and passed back half an eraser. “Yours is worn dull. Use this one.”

Lin Chen took it. Thanked him.

The boy lowered his voice. “I heard this month’s exam, the last physics problem is adapted from a provincial competition. Old Li wrote it. He likes testing critical points and extrema. That inclined-plane friction problem you worked out last night—you had the direction right. But the problem type may change. He’ll switch static friction to kinetic friction. Then add a spring.”

Lin Chen nodded. Said nothing. He put the half eraser into his pencil case, beside the half he already had. Two erasers. Different shapes. Same hardness. Enough. He opened the exercise book. Turned to the spring model. Drew the force diagram. Hooke’s law. F = kx. Equilibrium position. Maximum compression. He set up the equations. Checked them. Symbols. Units. No mistakes. He copied the steps into his wrong-problem notebook and wrote beside them: Critical point: velocity is zero. Acceleration is maximal. Spring force and the component of gravity are in balance.

At noon, the dismissal bell rang. The crowd surged toward the cafeteria. Lin Chen waited until the end. Avoided the crush. Walked to the long bench. Opened the oil-paper wrap. The rice was already completely cold. A hard crust had formed on top. He broke it apart. Ate it with the free soup. Swallowing mouthful after mouthful. The soup was clear broth. Two vegetable leaves floated in it. The salt taste was faint. He drank very slowly. Let the water enter his stomach fully. His stomach had something in it now. But not enough to stop the hunger. He put a hand on his abdomen. Pressed lightly. Digestive fluids secreted. The hunger receded. The cracks on the soles of his feet rubbed inside his shoes. Every step felt like stepping on broken glass. He shifted his center of gravity. Put his weight on the outer edge of his forefoot. Avoided the split skin. The edge of the sole had already peeled up. He pressed the curled part flat with a fingernail. It would not stay down. So he left it. As long as it did not affect walking. That was enough.

The ledger was in his canvas bag. He did not take it out. The figures were already in his mind. Balance: 1.56. Meal tickets for four days. Sixty cents a day. Four days, 2.40. That left 0.32. 0.32. Not enough to buy a new pencil. Not enough to buy a bottle of mercurochrome. Only enough for half a bar of soap. Or. Save it. For emergencies. He slipped his hand into his pocket. Touched the stub of candle. Two centimeters left. Burning time: about twenty minutes. He took it out. Set it in his palm. Weighed it. Light. But enough. Twenty minutes. Enough for one final major problem. Enough to check the arithmetic once. Enough to check the symbols once.

Afternoon class. Chemistry. Oxidation-reduction. Balancing equations. Lin Chen’s blind spot. He had never learned the electron-transfer method at the township school before. He only knew rote memorization. Now he had to make it up. He opened his notes. Broke the teacher’s method into three steps. Mark oxidation states. Find oxidation and reduction. Balance the charge. He practiced five problems on scratch paper. Got two wrong. Cause of error: forgot hydrogen ions in an acidic medium. He circled the mistakes and wrote beside them: Environment. Medium. H+/OH-. The handwriting was tiny. But clear. He was not seeking speed. He was seeking stability. Stable. That was speed. He copied the wrong problems into his notebook, marked the medium condition in red, the number of electrons transferred in blue, and the balancing coefficients in black. Three colors. Clear at a glance. Next time he met the same kind of problem, first check the medium. Then mark oxidation states. Finally balance it. The order could not be scrambled. Scramble it. And it would be wrong.

Evening study hall. 6:30 p.m. The classroom was full. The fluorescent tubes hummed. The air smelled of old books and sweat. Lin Chen finished that day’s homework. Closed his exercise book. Checked the time. 9:10 p.m. Twenty minutes until lights-out. He packed his schoolbag. The candle stub. Cardboard. Matches. Pencil. Eraser. Ruler. He counted them one by one. Put them into the canvas bag. Pulled the zipper shut. At 9:30, the lights-out bell rang. The hallway fell dark in an instant. Footsteps scattered. Sounds of washing up. Voices. Lin Chen waited. Counted to one hundred. The noise gradually thinned. He pushed the door open. Walked with his body close to the wall. The stairwell was between the fourth and fifth floors. The ventilation window faced north. Strong wind there. The rain had stopped. But the air was damp. He crouched down. Spread out the cardboard. Shielded the flame from the wind. Struck a match. Lit the candle. The flame flickered. Then steadied. He opened a real physics exam set. Timed. Forty-five minutes. Only the final major problems.

First problem. Conveyor belt model. Object starts from zero initial velocity. Coefficient of friction changes. Find the time. He drew a v-t graph. Segments. Accelerating. Constant speed. Decelerating. Set up equations. Substituted. Calculated t = 4.2 s. Second problem. Charged particle moving in a combined field. Electric field. Magnetic field. Gravity. Find the radius of the trajectory. He drew a force diagram. Lorentz force. Centripetal force. qvB = mv²/r. Solved for r. Third problem. A provincial-competition adaptation. Inclined plane. Pulley. Light rope. Critical angle. Find the mass ratio. He got stuck. The line of thought broke. He put down his pen. Closed his eyes. Breathed. In his mind he replayed what the physics teacher had said. Old Li liked testing critical points and extrema. He read the problem again. Looked for the invariant. Mechanical energy of the system was conserved. But friction did work. A non-conservative force. He changed approaches. Virtual work. No. Beyond the syllabus. He returned to the basics. Force analysis. Isolate one body. Treat the system as a whole. Set up the system of equations. Solve. Mass ratio m1/m2 = √3. He wrote down the answer. Checked the time. Forty-two minutes. Three minutes left. Check. Symbols. Units. Logic. No mistakes.

He blew out the candle. Packed his things. His fingers were numb with cold. His joints felt stiff. He hugged the canvas bag to his chest and stood up. His knees gave a faint crack. The pain in the soles of his feet had already dulled. Like a needle. Always piercing the ends of his nerves. He walked back down to the fourth floor step by step. Pushed the door open. The dormitory was very quiet. Only the even sound of breathing. He went to his bed. Sat down. Took off his shoes. Untied the gauze. The edges of the cracks had turned white. Tissue fluid was seeping out. No pus. He picked up the mercurochrome. A cotton swab. Dipped it. Applied it. A stinging pain. He held his breath. Spread it evenly. Wrapped it again. Every movement slow. But extremely steady. He lay down. Closed his eyes. In his head he was arranging time. Tomorrow. 6:30. Morning reading. 7:00. Breakfast. 8:00. Classes. 12:00. Noon break. 6:30 p.m. Evening study hall. 9:30 p.m. Back to the dorm. Wash up. Sleep. One step, one imprint.

He slipped a hand into the side pocket of the canvas bag. Touched last year’s real exam set. The edges of the paper had already curled. He took it out. By the moonlight outside the window, he looked at the blank space beside the last problem. The calculation steps he had worked through tonight. Filled it all up. But beside them was one line of small writing. Left by the boy in the front row. Note: When Old Li writes problems, he often takes the original model from issue no. 4, 1998, of Middle School Physics Teaching Reference.

Lin Chen stared at that line. His fingers tightened. 1998. Issue no. 4. The county library. Did not have it. The township school. Did not have it. He did not know where to look. But he knew this. Tomorrow, he had to make a trip to the county Xinhua Bookstore. Or. Find Old Li. Ask him. He closed the exam paper and put it back. The wind outside the window had stopped. The cloud cover broke apart. Moonlight shone across the cinder track. Puddles gave off a cold gleam. He placed his hands on his knees. Moved them slowly. The pain in the soles of his feet had already gone numb. His body felt hollowed out. But he did not sleep. He put his hands on his knees. Moved them slowly.

Tomorrow. 6:30. Morning reading.

He opened the ledger. The pencil moved.

Day 10. 22:10. Stairwell calculations.

Progress: 3 final major physics problems completed. 5 chemistry oxidation-reduction balancing problems.

Time spent: 45 minutes.

Status: On target.

Gap: issue no. 4, 1998, of Middle School Physics Teaching Reference.

Countermeasure: go to Xinhua Bookstore during tomorrow’s class break to consult it. If it cannot be borrowed, ask Old Li about the original problem model.

The pencil tip paused. He closed the ledger. His fingers tightened. The edges of the pages had curled a little. He shoved it into the very bottom of the canvas bag. Pinned it down. Left nothing showing.

Footsteps came from the corridor. Very light. Even rhythm. Not a student officer checking the dorms. A teacher on night rounds. The beam of a flashlight swept past the crack in the door. Stopped at the entrance.

“Sleep early. Don’t be late for morning reading tomorrow.” The voice was low and carried an echo.

“Got it,” the boy in the lower bunk answered vaguely.

Lin Chen said nothing. Only rested his hands on his knees. Moved them slowly.

The footsteps receded. The flashlight beam disappeared at the end of the corridor.

He opened his eyes. His gaze fell on the ledger by the bed. The pages glimmered pale white in the moonlight.

He reached out. Touched the pencil. Drew an extremely faint line across a blank page.

Day 10. End.

Progress: caught up three weeks. Blind spot: oxidation-reduction medium. Funds: 1.56.

Status: alive.

The pencil tip paused. He set the pencil down. Closed his eyes.

Tomorrow. 6:30. Morning reading. Xinhua Bookstore. Opens at 8:00. Ten-minute class break. Distance: 800 meters. Walking. Eight minutes. Consult: two minutes. Return. Just enough. He could not be late. Could not be discovered. Could not overspend.

He placed his hands on his knees. Moved them slowly. The pain in the soles of his feet had already gone numb. His body felt hollowed out. But he did not sleep. He put his hands on his knees. Moved them slowly.

Outside the window, the sky began to darken. The clouds hung very low. In the distance the chimney was breathing white smoke. The wind carried the damp, earthy smell of wet soil.

Wind was coming.

He slipped a hand into the pocket inside his clothes. Touched the notice stamped with a red seal. The paper was already warm. Its edges had curled a little. He took it out. Opened it. Read it through carefully once.

Report in. 8:00 a.m. Assemble at the township education group. Travel together by vehicle to the county.

The time had passed. But the rules were still there. One step, one imprint. No disorder.

He folded the notice. Put it back. His fingers tightened. The edge of the cloth bag pressed into his palm.

He closed his eyes. His breathing was steady. In his head he was arranging time. Tomorrow. 6:30. Morning reading. 7:00. Breakfast. 8:00. Classes. 12:00. Noon break. 6:30 p.m. Evening study hall. 9:30 p.m. Back to the dorm. Wash up. Sleep.

One step, one imprint.