The chamber's collapse slowed as Mark gripped his chalk, the shadow's words echoing: Give us a piece of yourself. The air was thick, pressing against his chest, and the glowing equation—4^x = 64—pulsed like a heartbeat. Lyra's hand was still on his arm, her grip tight, her eyes wide with something like fear.
"Don't give it anything," she said, voice low. "The Tower takes more than you offer."
Mark's mind spun. He'd solved the equation—x = 3—so why wasn't the arch opening? The shadow loomed closer, its form shifting between human and something else, its eyes a whirl of numbers. "Choose," it said. "A memory, a fear, or a hope. One piece, and the path continues."
Mark's throat tightened. A memory—his mother's voice, teaching him to count by candlelight. A fear—failing Numina's people, who whispered his name in the streets. A hope—to master the Tower and reshape the world. Each felt like a piece of his soul.
Lyra's voice cut through. "It's a trick, Mark. The Tower doesn't need your answer—it needs your will. Fight it."
He nodded, heart pounding, and raised his chalk. "I've solved your equation," he said to the shadow. "x = 3. That's enough."
The shadow laughed, a sound like breaking glass. "Bold. But numbers alone won't save you." It raised a hand, and a new problem burned into the air:
Solve: √(81) + 3²
Mark's mind steadied. Square roots and exponents—simple, grounding. √81 = 9, and 3² = 9. So, 9 + 9 = 18.
"Eighteen," he said, staring down the shadow.
The chamber stilled. The shadow's eyes flickered, then dimmed. The gears stopped grinding, and a new archway opened, its edges soft, glowing green. Numen flooded Mark's Seal, warm and alive, but heavier than before, like carrying a storm in his veins.
Lyra let go of his arm. "You're stubborn," she said, a faint smile breaking through. "Good. You'll need that."
The Green Arch: The Logic of Roots
The new arch led to a garden-like chamber, its walls overgrown with vines that pulsed with numbers. The air smelled of earth and ozone, and a voice—gentle, almost kind—spoke:
"To grow, you must find the root. Solve: √(x) = 5"
Mark exhaled. Square both sides: x = 5² = 25.
"Twenty-five," he said.
The vines parted, revealing a path lined with glowing stones, each etched with a number. But as he stepped forward, Lyra grabbed his sleeve. "Wait. Look."
One stone was different, pulsing red, with a new problem:
Solve: (x + 3)² = 16
Mark worked it through. Square root both sides: x + 3 = ±4. So, x = 1 or x = -7. Two solutions again. He remembered the Obsidian Arch, the weight of choosing -4. "One," he said, picking the positive.
The stone turned green, and the path extended, but Lyra's face was grim. "You keep choosing the light," she said. "But the Tower loves balance. Negative numbers, shadows—they're part of it. Ignore them, and you'll pay."
"What do you know about it?" Mark snapped, frustration bubbling. "You keep warning me, but you're not saying anything."
Lyra's eyes flashed. "I've seen people walk this path. My brother did. He chose wrong, and the Tower took him. That's all you need to know for now."
The Final Test
The path ended at a massive door, its surface a mosaic of numbers and symbols. A final problem appeared, carved in gold:
Solve: 3x² – 12 = 0
Mark's hands shook. He factored: 3(x² – 4) = 0, so x² = 4, and x = ±2. Another choice. Lyra's words echoed—the Tower loves balance. He couldn't keep dodging the negative.
"x = -2," he said.
The door blazed, and Numen surged into his Seal, sharp and cold, but stronger than ever. The mosaic parted, revealing a blinding light and a figure beyond—a scholar in a robe of shifting numbers, holding a staff that hummed with power.
"Welcome, Mark of Numina," the scholar said. "You've passed the trials of logic. But the true test begins now. The Tower has chosen you—and another."
Lyra stiffened beside him. The scholar's gaze flicked to her, then back to Mark. "Two Scholars. One destiny. Only one can claim the Heart of Numbers."
Mark's stomach dropped. Lyra's hand went to her Seal, her face unreadable. The scholar raised his staff, and the light consumed them.