Share Them Again.

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

Saeon reached the table half a step ahead. Three silver trays sat on the eight-sided navy stone top. Same rounded corners. Same size.

The three little lamps around the rim were dark. Dure set the notebook on the side shelf. The broad hall stayed open behind them.

The teacher placed the cores in the middle. Twelve round spheres glowing blue. As the teacher's hand drew back, three clear rows appeared.

“Today, we fix a method. Twelve cores. Equal shares for three.”

Saeon followed a row of four with their eyes. The same total as the first session. The trays that had held fifteen last time were empty.

Saeon, Sodam, and Dure would share the twelve cores. Saeon rested a hand on the nearest tray. Again, they needed to find how many one person would get.

“I found three methods.”

Ichan spoke from upstairs, tapping the railing from safely behind it. Saeon looked up.

Last time, Saeon had said three next time. But the empty trays suggested something to do first. Before finding more methods, Saeon wanted to show one from start to finish.

“Three can wait. Today, one that works.”

Ichan lowered the hand and waited. Saeon looked back at the table and tapped the empty tray's edge.

“I want to lay it out first.”

Maybe putting twelve and three separately in front of them would help. Saeon pushed three cores forward from the middle. They stopped in a row just inside the table's edge.

Nine remained in the middle. A small gap separated them from the three set aside. Saeon opened both hands and studied the groups.

Take three away from twelve, and nine remained. Three people were sharing. Did that mean taking away these three? Saeon's eyes stayed on the nine.

“So the nine left over is one person's share?”

Sodam looked at the row in front, adjusted the glasses, and lowered the hand.

“Done? Have you checked everything?”

Saeon reached toward the nine, then stopped just before moving them to the nearest tray. Sodam's and Dure's trays were still empty.

If one person got nine, where would the other two get theirs?

Saeon lowered the hand. The three cores set aside were still glowing blue. They were part of what needed sharing, but Saeon hadn't decided who would get them.

An amber knot appeared on the right-hand lock. Dull strands of light formed rounded layers, with two cream-colored gaps like eyes.

The knot wrapped around the right lamp, too. It stayed within a space about as big as one tray. It didn't reach toward the children or block the hall.

Through the outer strands, a smooth inner layer showed. It had the same tidy shine as the Mixed-Up Places Tangle. The lamp beneath it stayed dark.

The teacher pointed to the wrapped objects.

“A device made from a mistaken idea. Two layers this time. Explain, and it'll come undone.”

Saeon looked from the three cores in front to the others. Setting aside three spheres didn't change the number of people. Sodam and Dure still stood at their trays.

“Three was the number of people.”

Saeon's fingertip moved from the nearest place to Dure's, then to the spheres pushed forward. Those three weren't objects to take away in place of people.

“These are cores we need to share, too.”

Sodam pointed beside the row set apart.

“None left over. Those three count.”

Saeon returned the cores to the middle. Three rejoined nine. All twelve lay between the trays again.

“Put them all together. One each till the middle's empty.”

One on the nearest tray. One for Sodam. One for Dure. After a full round, each person had one.

Nine cores remained in the middle. The same number as before. But this time, there was one on every tray.

Saeon pictured the two arrangements side by side. Before, three had simply been set aside. Now nine remained because three people had each received one.

“Nine is what's still in the middle.”

Saeon turned to Sodam and added,

“Not what one person got.”

Sodam pointed to each tray. Saeon waited until the fingertip reached Dure's. Next round, everyone would get the same amount again.

One here, one for Sodam, one for Dure. Two on every tray. Six in the middle.

Saeon paused before reaching back in. No spheres waited along the front rim, either. Only the six in the middle remained to be shared.

Another round made three on every tray. Three stayed in the middle. Saeon pointed to the nearest tray with an empty hand.

Last time, this was where Saeon had called it done. Three people and three cores each had seemed to match. Today, there were three in the middle as well.

“Three people. Three each so far.”

Saeon pointed to the two places separately. The smooth inner layer still covered the lamp. The same number appeared, but it meant different things in different places.

“Some still in the middle. We're not done.”

Saeon took the final round slowly. One here, one for Sodam, one for Dure. The last core went down, and Saeon opened both hands.

Four on all three trays. The middle and front rim were empty. No spheres in Saeon's hands, either.

Then the shallow recess at the front opened. Sodam pointed from outside it.

“There's another piece.”

Saeon pulled both hands back and leaned closer. Inside lay the rounded, three-step stone piece from the last math session.

The gap under its lowest step was still there. Just below it lay a smaller stone piece. Exactly the same shape as the gap.

A line of light ran down the stairs and reached the smaller piece. Saeon studied the space between them. The light touched it. The stones were still apart.

No letters or patterns anywhere. The smaller piece lay by itself on the floor of the recess. Nobody reached in or took it out.

Saeon remembered the upper hollow from the last science session. There, light had traveled upward. Here, a new piece lay below the gap underneath.

The recess slowly closed. Saeon straightened. Four cores still sat on every tray. The lamps remained dark.

The outer strands had loosened. The inner layer still shone smoothly. Saeon looked first at the three trays and the empty middle.

“Three is the number of people. Four is each person's share.”

Saeon pointed to the place where the three cores had been pushed aside. At first, only twelve and three had mattered. The places needing equal shares had been forgotten.

“If we just take three away, we miss the ‘equal shares’ part.”

Sodam lowered a hand from the glasses.

“That's my way of checking.”

Saeon looked at Sodam, then pointed to the empty middle again. Even beneath that tidy shine, there was something to check. This time, Saeon hadn't waited for a friend to say it first.

A thin cream-colored pin of light slipped from Saeon's fingertip. It stretched over the empty middle and stopped at the smooth layer around the right lamp.

Saeon's hand didn't touch the knot. The pin didn't hide the cores, either. The smooth amber shine stilled, revealing the separate strands beneath it.

Dure ran a fingertip along the nearest tray. After checking the four cores, Dure looked at the middle.

“Equal shares. Empty middle. Done.”

Both layers broke into cream-colored specks. The lock and the right lamp emerged. The blue cores stayed on their trays.

All three lamps came on together. Saeon looked across the trays of four and drew back the hand above the empty middle.

Twelve shared equally among three made four each. A short equation appeared beside the table.

Sodam pointed from one tray to the next.

“Check it backward. Three lots of four make twelve.”

Saeon added four and four to get eight. Add Dure's four, and it was twelve. Every sphere from the start was now on a tray.

Saeon looked at the teacher.

“Teacher Gwak, I spotted it this time.”

Saeon opened a hand toward the place where the cores had been set aside.

“Three meant people. It didn't mean taking three cores away.”

The teacher waited for Saeon to finish, then nodded.

“Nice and clear. Both layers.”

“You got it. Three methods at the move-up challenge.”

Ichan finished speaking from behind the railing. Saeon looked at the two friends beside the glowing trays. In every round today, all three had a place.

“And all three of us will explain.”

Dure fetched the notebook from the shelf. Sodam lifted a hand from the tray. Saeon joined them in the open hall.

Still the bottom floor. Saeon looked up, remembering the little gap in the recess. A stone with the very same shape lay just beneath it.

What was at the top?