On the Trays.

At the next math practice, Saeon looked at the tabletop first. Last science practice, they had watched two ends. Saeon remembered the magnets pushing apart even when brought close.

Sodam stood beside Saeon. Dure hugged the notebook. Teacher Gwak Doeun was there, too.

Blue light glowed over the eight-sided, dark blue stone table. Saeon stepped half a pace ahead. Three familiar silver trays stood in a row.

Dure set the notebook on the shelf beside them. A wide hallway lay open behind the three friends. All three little lamps around the table were off.

“They're in groups of twelve. How many altogether?”

The teacher moved an empty hand away from the table. Saeon studied the cores on their tray. Round, smooth, glowing blue balls.

Four made one row. There were three rows. Add four, then four again, then four more. Twelve.

Saeon had one group. Sodam's tray in the middle held one, too. Dure's tray on the right held two.

There was a gap between those two groups. Each had three rows of four. All three trays were the same size.

The middle of the table was empty. No cores had been set aside. Saeon straightened a little to see all three trays at once.

“I've got another way. For today's problem.”

Ichan spoke from the floor above. Standing straight behind the railing, Ichan opened one hand. The hand stayed inside the railing.

Saeon looked up. Last math practice, they had asked Ichan to show another way. Today, Saeon wanted to find it first.

“We'll find it and explain it ourselves.”

Ichan lowered the hand and waited. Saeon tapped the edge of their tray without touching a ball.

“My tray has twelve.”

Saeon's finger was already heading toward the middle. Last time, every tray had held thirteen. They could point to each tray and add.

There were three trays today, too. Sodam followed Saeon's hand, resting a finger on the middle tray's edge.

“Three trays. Twelve each. Is that it?”

Saeon glanced toward Dure. Sodam adjusted the arm of their glasses, lowered the hand, and spoke.

“Twelve three times is thirty-six.”

Sodam's finger stopped in front of the right tray. Balls lay on both sides of the gap. Saeon pulled back the hand that had been about to point.

Had they counted the group beyond that gap?

Dure brought their fingertips together in front of the tray.

“My tray has two groups.”

Saeon looked from left to right. The blue rows inside those matching silver edges were different. Saeon had three rows. Dure had two groups of three rows.

Amber light wrapped around the lock on the right. Cloudy strands twisted into a round knot, smaller than one tray.

Two cream-colored gaps opened between the strands, like eyes. An amber coating covered the right lamp beside it. Its smooth surface made the lamp look lit.

Saeon checked the left lamp. The front lamp was still off, too. Only the right one had this different light over it.

“Is the right one already on?”

Saeon tapped the table edge in front of their tray. The knot twitched once in place. The trays stood between Saeon's hand and the knot.

The strands stayed around the lock and lamp. They did not come toward the children. The hallway behind them stayed open.

“Hold on. A thought got tangled.”

The teacher gestured toward the two objects.

“It's a device made from a mistaken idea. Each of you, explain why.”

Saeon wanted to split up Dure's cores and share them with their own tray. Then the trays might have the same amount. But Saeon's hands stayed outside the edges.

Today's rule was to leave the groups alone. They did not need to make equal shares, as in their first practice. They needed to count all the cores already there.

Sodam looked down at the middle tray again.

“I counted the trays. I treated each one as twelve.”

Sodam pointed to the gap on the right tray, without moving or covering any balls. Saeon followed the fingertip.

“There are two twelves here.”

Sodam's finger stopped below the group to the left of the gap. Then it moved below the group on the right. Two stops inside the same tray.

“The trays have different amounts. But there are four groups of twelve.”

Saeon looked at the silver edge they had pointed to. One edge did not always mean twelve. They had to look at the groups inside.

Sodam started again in front of Saeon, tracing outside each group instead of pointing to individual cores.

“Twelve.”

Sodam pointed to the middle tray.

“Add twelve. Twenty-four.”

The finger moved to the first group on the right tray.

“Twelve more. Thirty-six.”

This time, the hand did not stop at the tray's edge. It crossed the gap and went below the second group. Saeon looked at those three rows again.

“One more twelve. Forty-eight.”

Sodam drew back the hand and studied all four groups.

“Four groups of twelve. Forty-eight.”

Saeon pointed to the two groups on the right separately, then checked the middle and left. Every core was still where it had started.

“Three trays, but four groups.”

A shallow recess opened in the front of the table. Dure swept a hand along the edge in front of their tray, then stopped to look down.

“The empty slot at the top filled with light.”

Saeon leaned forward a little. Inside lay three stone steps with rounded corners. A small stone sat separately below the lowest step.

The space between the empty place below and that little stone was unchanged. Saeon could see the upper slot from last math practice. It was shaped like one step, just above the highest step.

Last time, the tip of the light had reached it. Today, blue light filled it right to the edges. There was no stone piece inside the light.

Saeon shifted their eyes to the side. The bottom of the bright slot was visible. Light had filled it. The place for a stone was still empty.

“Still no piece.”

Sodam lowered a hand outside the recess. No letters or patterns appeared. None of them touched the little stone.

The light had climbed higher during science practice, too. But that upper slot had been empty. Saeon wished they could see the light in both places side by side.

Instead of a piece, Saeon memorized the edges in front of them. The recess slowly closed. When Saeon straightened, the right lamp came into view again.

The amber coating was still there. This time, Saeon would not call it finished just because of that shine. First, they had to explain how they had counted every group.

Sodam's fingertip moved over the bare stone between the groups. A long, thin, cream-colored pin stretched out. It reached the knot on the right without covering any cores.

Sodam's hand never touched the knot. The strands of light stopped moving. Dure swept a fingertip once along the edge in front of their tray.

It stopped below the second group.

“Twelve in each group. Four groups. We've counted them all. Finished.”

The knot loosened into cream-colored specks. The smooth coating peeled off the right lamp. No strands remained on the lock.

All three lamps came on together. This time, the right lamp's light matched the others. Saeon checked the lights, then turned back to the cores.

They had added equal twelves and counted all four groups. No cores were left in the middle or in anyone's hand. No group had been split or joined.

A short equation appeared beside the table.

Multiplication showed the same number added several times. In the twelve, Saeon saw one group. In the four, they saw all four groups across the trays.

“Four groups of twelve. Forty-eight altogether.”

At first, Saeon had thought three taps, one per tray, would finish it. Now there were different places for a finger to stop, even inside one tray. Saeon checked Dure's again.

Twelve in the left group. Twelve in the right. Together, Dure's tray held twenty-four. Sodam traced beneath both groups.

“If we add by trays, this one's twenty-four.”

Saeon started with their own tray again.

“Twelve, twelve, twenty-four. Add those, and it's still forty-eight.”

Counting four groups counted the same cores. Adding by trays left no group out, either. Dure pulled back an empty hand.

“Both ways counted everything.”

Saeon looked at the teacher.

“We found two ways!”

Sodam looked up, too.

“We counted the same-sized group four times. For the tray way, we checked how many were in each tray.”

The teacher waited until all three had finished, then nodded.

“Nice and clear. Both ways.”

Saeon opened a hand above the empty middle. Last time, they had pointed to one tray at a time. Today, they had found equal groups even in different places.

“You found that way.”

Ichan spoke from behind the railing. Saeon looked up. Ichan opened a palm and went on.

“I'll bring more ways to the move-up challenge.”

“Then we'll find more, too.”

Saeon turned toward Sodam and Dure. Sodam lowered a hand from the table. Dure took the notebook from the shelf and hugged it.

The three friends left the glowing table for the open hallway. They were still on the lowest floor. As they walked, Saeon looked up.

That little upper slot was full of light, even without a stone. Saeon pictured the empty place, then the tower stairs. The unseen top was still far away.

What was at the top?