Keep the Groups Together.

At the next math session, Saeon walked to the Gallery. Sodam kept pace alongside, and Dure hugged the notebook. Teacher Gwak came with them.

The eight-sided navy stone table came into view. The same three silver trays stood in their usual places. Saeon moved half a step ahead, then stopped.

Today, the trays weren't empty.

Round blue-glowing cores filled them. None lay in the middle. The teacher stood with both hands away from the trays.

Dure put the notebook on the side shelf. The broad hall stayed open behind them. All three round lamps along the table's edge were dark.

“Today, we're working with groups. How many altogether?”

Saeon looked down at the nearest tray. Five in the first row. Five in the next. Three in the last, shorter row.

Five and five made ten. Three more made thirteen. None of the spheres overlapped, so every one could be counted.

“Thirteen here. Next.”

Saeon tapped the tray's edge, already looking toward Sodam's. Five, five, three there, too.

The same rows sat in front of Dure. Thirteen on every tray. The cores were all the same size and glowed the same blue.

Last time, they had shared out the spheres in the middle. Today, they needed the total already on the trays. Saeon looked at the rims instead of the cores.

“There are three methods for these groups, too.”

Ichan spoke from upstairs, standing straight behind the railing and tapping its top lightly.

Saeon looked up. That promise from the last math session came back. Even if they found more methods, Saeon couldn't do all the talking.

“All three of us will explain.”

Ichan lowered the hand and waited. Saeon looked back at the nearest tray. Thirteen was already settled.

That left three trays. Saeon raised a finger above the empty middle. Point to each tray, and this could be quick.

“Thirteen plus three. Done?”

Sodam looked at Saeon's hand, then down at the tray.

“Let me look, too.”

Sodam hadn't said right or wrong yet. Saeon stopped before pointing across the trays. The fingertip hovered over the empty middle.

Did pointing to one tray mean counting one core?

Saeon drew back. Counting thirteen had meant looking at each sphere. The three Saeon was about to add didn't stand for spheres at all.

Amber light gathered around the right-hand lock. Dull strands made a small, round knot. A Tangle smaller than a tray.

Two cream-colored gaps like eyes showed between the strands. The knot wrapped only around the lock. The right lamp beside it remained uncovered.

Saeon tapped the rim in front of the nearest tray. The knot gave a little twitch in place. Its strands didn't reach toward the children.

“Wait. That idea got tangled.”

The teacher pointed to the lock.

“A device made from a mistaken idea. Each of you, explain why.”

Saeon almost moved a tray, then lowered the hand. Gathering everything in one place might make counting easier. But Sodam pointed to the edge, not the spheres.

“Keep the groups together. One tray at a time.”

Sodam pointed outside Saeon's tray and followed the rows without hiding any cores. The look stopped at the short final row.

“Thirteen here.”

Saeon checked those thirteen again. Sodam's fingertip moved to the middle tray. Thirteen there, too.

“Thirteen here, too. Add them. Twenty-six.”

Saeon looked at the two trays side by side. Sodam had pointed to just one more tray. But the number added was thirteen, not one.

The short row on Dure's tray still waited. Sodam pointed to its rim. Saeon opened their mouth, closed it, and waited.

“Add thirteen again. Thirty-nine.”

Sodam traced the outside of the three trays in order.

“Thirteen. Twenty-six. Thirty-nine.”

Not a core had moved. All their hands were empty. They had added the same thirteen once for each tray.

Then a shallow recess opened at the front. Dure drew the fingertips together and looked inside. Saeon bent closer without reaching in.

“An empty hollow at the top.”

Inside lay the rounded three-step stone piece. The gap beneath its lowest step showed, too. The smaller stone still lay separately just below.

Saeon found the space between them from the last math session. The little stone and the stairs still hadn't joined. This time, Saeon's eyes traveled farther up.

Just above the top step was an empty hollow shaped like one more step. A line of light climbed the stairs and reached the empty space.

No letters or patterns. The light reached it, but there was no stone to fill it. Saeon kept both hands outside.

The hollow from science was still empty, too. But they had never put that piece beside this one. Saeon studied only the empty space actually in front of them.

“The piece underneath hasn't moved, either.”

Sodam pointed downward from outside the recess. Dure looked at the upper part and lowered the hand. Nobody touched a piece or took one out.

The recess slowly closed. Saeon straightened. Strands still clung to the right-hand lock.

All three lamps stayed dark. Seeing a clue hadn't explained Saeon's own reasoning. Saeon looked at the trays that the pointing finger had been heading toward.

“After thirteen, I was going to count the trays one by one.”

Sodam adjusted the glasses, lowered the hand, and pointed separately to the trays and cores.

“Three is the number of trays.”

Then Sodam pointed to the three rows on the nearest one.

“Thirteen is how many are in one tray.”

Saeon followed each gesture. A tray wasn't one loose sphere. It held thirteen that needed to be counted together.

Sodam's fingertip moved toward Dure's tray.

“Not thirteen plus three. Three groups of thirteen.”

Saeon nodded. Thirteen plus three would make sixteen. But right here were three trays, each holding thirteen.

“I treated a whole tray like one core.”

Saeon pointed from the nearest tray to Sodam's.

“I need to add thirteen again.”

A long, thin cream-colored pin of light came from Sodam's fingertip. It passed over the empty middle and stopped at the knot on the right, leaving the cores in view.

Sodam's hand didn't touch the knot. The amber strands stilled in place. Saeon turned toward Dure.

Dure brushed a fingertip along the nearest tray. After checking the last sphere, the finger rested at the end of the rim.

“Three trays. We added them all.”

Dure pointed from Saeon's tray across to the last one.

“Thirteen in each. No tray left out. Done.”

The last strand broke into cream-colored specks. The lock emerged, and all three lamps came on together.

Saeon checked the trays again. Still thirteen each, just as before. No cores set aside in the middle or held in anyone's hands.

Same number in every tray. All three trays added. Saeon could see both what Sodam had checked and where Dure had finished. Thirty-nine altogether.

Short equations appeared beside the table.

Multiplication showed the same number being added several times. Saeon looked from thirteen in the equation to the nearest tray. At three, Saeon looked across all the trays.

“Thirteen each. Three trays. Thirty-nine altogether.”

Saeon looked up at the teacher.

“Teacher Gwak, Sodam found it this time.”

Saeon looked at the rims Sodam had pointed along.

“We added one tray at a time. We didn't move any cores.”

Sodam looked at the teacher, too.

“We added the same thirteen three times.”

Dure lifted the hand from the end of the tray.

“We stopped once we'd added all three.”

The teacher waited until everyone had finished, then nodded.

“Nice and clear. All three of you.”

“You got it. I've got another method.”

Ichan spoke from behind the upper railing. Saeon looked up and raised a hand a little. First, they had finished putting today's method into their own words.

“Show us. We'll explain it our way.”

Sodam drew back from the table. Dure fetched the notebook and closed it. They left the glowing lamps and stepped into the open hall.

Still the lowest floor. As they walked, Saeon looked up. The light in the stair piece had reached the upper hollow. But the empty space remained.

What was at the top?