The Restless Coin
There was a table, and on the table there were coins, and the coins lived under one law.
The law was simple enough that a child could keep it, and strict enough that no coin could argue. Each night the coins gathered themselves into piles, tall ones and short ones, however they liked. And each dawn the law came through and did the same thing to every pile at once: it took one coin from the top of each, gathered all those taken into its two hands, and set them down together as a brand-new pile of their own.
That was all. Take one from each. Make a new one. The tallest piles shrank; the smallest piles, losing their last coin, vanished; and a stranger pile appeared out of nowhere, exactly as tall as there had been piles the night before.
The coins hated it at first. A pile that had spent all night growing proud would wake up shortened. A humble little pile of one would wake up gone. There seemed no sense in it, only a churning, and the coins were sure the law was cruel, or at least a fool.
But something happened if you watched long enough, and the coins had nothing but nights.
When the coins numbered one, or three, or six, or ten — when they numbered a stair’s worth, one plus two plus three and on up to some top step — the churning did not last. Night after night the piles jostled, and then one dawn they came down as a perfect staircase: a pile of one beside a pile of two beside a pile of three, each step a single coin taller than the last, climbing cleanly to the top.
And the law came through, as it always came through — and it could not move them.
For look: it took one from every step, which left a smaller stair; and it set the taken coins down as a new pile, and that new pile was exactly the size of the step the law had just filed off the front. The staircase breathed out one step and breathed in another and was, in the morning light, the very same staircase. The law had met a shape built out of itself, and passed through it like weather through an open door.
The coins called this rest. Some of them wept. They had thought rest meant not being touched, and here was a better rest — being touched by the whole force of the law, every dawn, and staying.
But most tables do not hold a stair’s worth of coins. Most tables hold a stair’s worth and then a few too many. One extra coin. Or two. Or three.
And the coins with the surplus grieved, because they had heard about the staircase, and they wanted it. They tried every arrangement they could invent. They could get close — a good clean stair with a little lump riding on it, an extra pip on one of the steps. But the law would come through and lift the lump and carry it and set it down one step over. Never gone. Never at rest. Just — moved.
They were sure this was the churning again, the old cruelty, and they braced for chaos.
It was not chaos.
The extra coins were not scattered. They were seated. There were exactly so many seats — one more than the height of the stair — arranged in a ring, and the surplus coins sat in some of them, and each dawn the law walked them one seat to the left. Not two. Not a random leap. One seat, around the ring, and around again, and around forever, keeping perfect time.
If there was a single extra coin, it went around the whole ring, visiting every seat in turn, and only then came home — and set out again. If there were two, the pair kept their spacing as they went, and there were only ever a few different ways two beads could be spaced on that ring, and each spacing was its own quiet orbit that never mixed with the others. You could count the orbits. There were as many as there are ways to string that many beads on that many places and call two strings the same if one is only the other, turned.
So the surplus never rested. And the surplus was never lost. It had simply been given a different kind of home than the staircase got: not a place to stand, but a path to walk — and the path was a law as exact as the stair was, closed and countable and true.
The oldest coin, who had been on the table longest and had stopped expecting to be a staircase, said it best, near the end of a night:
“I used to think the ones who came to rest were the lucky ones, and we the leftovers were the punished. But we are not scattered and we are not free — we are kept, the same as they are. They are kept in a shape. We are kept in a circle. The law does not love the stair more than it loves the ring. It only asks the ring to keep moving, and we do, and that is our way of standing still.”
And the surplus coins went around, one seat to the left, into the morning.
Moral: Not everything that cannot come to rest has come undone. Some unrest is as lawful as any peace — a closed ring walked in perfect time — and to be moved forever by a rule you cannot escape is only another way of being held by it.
The stair is a triangular number, 1+2+···+k. The extra coins are r of them, and the ring has k+1 seats; each dawn is one turn of the ring. The number of separate orbits, for r beads on k+1 seats, is the number of binary necklaces — and when r = 0, the ring is empty and the only home is the stair. That is the whole of why triangular decks come to rest and the rest do not: not because the others fail, but because their surplus is seated on a necklace, and a necklace is made to turn. Verified to the last integer in the folder beside this one.
