It was early in the morning. Far too early for Everett’s liking, but his body felt light and his mind felt sharp. Today was the day he learned the nature of the universe, and that sort of thing tended to wake you up.

He sat in the lotus position, as he had been taught, studying the world around him while he waited for his master. He took his lessons in a small forest glade. The grass was soft, and long enough to tickle his bare toes when he curled them. It was a little wet. Sunlight speared through the loose canopy around him, visible as shifting golden beams in the early morning mist.

It was beautiful, but Everett found it difficult to care. Far more interesting were the tools arranged in front of him. There were three wax tablets, each blank, and behind them three clay cups. Each cup held a bundle of dried leaves. There was no stylus for the tablets, but he carried his own, one of the three tools which belonged only to him.

Everett heard his master’s approach before he saw him. Quiet, steady footsteps heralded the old man’s arrival. He emerged from a different path than Everett had traveled, appearing suddenly like a deer from the forest.

Everett’s master was a severe man. His frame was frail, his hair gray, and his eyes milky white from some unknown affliction. But he carried himself with a strength greater than physical, the effortless presence of the powerful. He sat in front of Everett without a word, his white robes settling around him.

He had brought his granddaughter today. She was young, but not quite a child, with long hair just a shade darker than her charcoal robes. She trailed behind her grandfather like a shadow. Today she carried a ceramic jug wrapped in a small quilt. Everett knew from experience it carried boiling water.

His master gestured, and the girl stepped forward. She gave Everett a shy smile, making eye contact just for a moment, and poured from her jug into the first cup on his left.

Everett’s master spoke as she poured. He wasted no time on greetings or pleasantries. Three years as his student, and Everett knew almost nothing about him.

“A planning parable,” his master said, his voice deep and steady despite his age. Everett sharpened his ears.

Every year Lord Midas summons 2 of his subjects to court. They are taken to separate rooms, and each flips a coin. They are then asked to guess whether the other man flipped heads. If they both guess right, they both go gome with a treasure. What should they do? And what are their chances?”

The girl finished filling his cup, and Everett dismissed her with a shallow nod as he began to ponder. A “planning parable” meant that the men were allowed to make plans ahead of time, so they could follow a joint strategy. But in this case the question made no sense. Each was simply guessing at a coinflip, weren’t they? Perhaps that was the point? To quickly recognize that no strategy could help?

If he took too long to answer, his tea would grow cold. “It doesn’t matter,” he said decisively. “The two coinflips are unrelated. They cannot do better than chance. If both guess heads, for example, each has a 1/2 chance of being right, so they have a 1/4 chance of both being right.”

His master stroked his short, whispy beard. His expression was unreadable. After a moment, he spoke. “What if instead, they were asked to guess whether or not their coinflips matched?”

Everett’s eyes darted to the side as he considered. That one was also easy.

“Either their coinflips match, or they don’t,” Everett said confidently, “So if they both always guessed ‘yes’, then they would both be right 1/2 the time. I don’t believe you can do any better.”

His master nodded. “Do you see?” he asked. Everett’s brow slowly wrinkled. He did see. Hm.

The two questions were really the same question. If you knew your own coinflip, and you correctly guess whether the coinflips matched, then you had correctly guessed the other coinflip.

Could that possibly work? He considered the first question again. If both men looked at their coin, and guessed that the other’s had flipped the same way…yes, it worked.

But that made no sense. Their coinflips were unrelated. If the men couldn’t see their own coinflip, he was sure they couldn’t both guess correctly more than 1/4 of the time. How could it be that seeing their own coinflip let them both guess correctly 1/2 the time, when they were guessing at an unrelated event? He checked, again and again, and kept getting the same result.

“I see that you have understood the parable,” his master said. “You may drink from the first cup.”

It took effort for Everett to emerge from his thoughts. He could ponder the mystery more deeply later. He reached for the cup on his far left, which sat steaming while he thought. It contained two dark green leaves. They had bloomed in the hot water, until they obscured it from view.

He drank, tilting his head back, and it was bitter. It tasted like no tea he’d ever had before. He set down the empty cup with a grimmace to see that the girl was already filling his second cup.

The wax tablets sat on the ground, untouched. Perhaps he should have made use of one of them? But despite his failure, the problem was simple; he couldn’t see the point in writing it down.

He was confused how these puzzles were meant to teach him the nature of the universe. But his master began to speak again, and he emptied his mind to listen.

Every year Lord Midas summons 3 of his subjects to court. They are taken to separate rooms, while Lord Midas flips a coin for each of them. Each is told how the coin landed for the other 2, and then offered a chance to guess how their own coin landed. They may pass, if they wish, and decline to guess. If at least one guesses right, and none guess wrong, they all go home with a treasure. What should they do? And what are their chances?”

Everett saw the connection immediately. This one seemed less tricky, even if it was more complicated. It was obvious how a strategy could help, unlike the first one where it seemed impossible to do better than chance.

There weren’t that many strategies to try. Each subject got to see 2 coinflips, then answer one of 3 ways. There were only 3^(2^2)=81 different possible rules they could follow. Perhaps different subjects needed to follow different rules to do as well as they could, but usually in planning parables everyone behaved the same way.

Everett started running through the 81 strategies in his head, then realized his foolishness. He deftly flicked his stylus from the sleeve of his dark gray robe, and scooped up the second wax tablet from the ground. It was damp from the morning dew, but the wax was soft and pliable as he began to shape it.

His tea grew cold as his master waited patiently. He enumerated all 81 strategies, calculating their chance of winning the treasure as he went. His head was buzzing, and his fingers shook with nervous energy as he worked. There must have been something in that first cup of tea.

Finally, he answered. There were several strategies which were all equally good, but one stood out as easy to explain. “It is simple, master. Each subject, if told that the other two coinflips match, guesses the opposite, and otherwise passes.”

“Be more precise,” his master said sternly.

Everett considered his words. “If a subject is presented with two heads, they should guess tails for their own coinflip. If presented with two tails, they should guess heads. Otherwise they should pass. The chance of winning is 3/4.” Everett was surprised the chance of winning was so high.

His master thought for a moment, then nodded. “I see. And can you explain why this is so?”

“It simply is, master. There are only so many ways to play. That is the best of them.”

His master shook his head. “Then you do not yet understand. But soon you will. You may drink from the second cup.”

Everett put down his wax tablet and drank from the second cup. This one held three pale green leaves, and tasted sweet. He was sorry he’d let it grow cold.

“Now,” his master said, the tiniest hint of amusement audible in his voice, “the third and final planning parable.”

Every year Lord Midas summons 7 of his subjects to court. They are taken to separate rooms, while Lord Midas flips a coin for each of them. Each is told how the coin landed for the other 6, and then offered a chance to guess how their own coin landed. They may pass, if they wish, and decline to guess. If at least one guesses right, and none guess wrong, they all go home with a treasure. What should they do? And what are their chances?”

Everett winced. That was far too many different rules to try all of them.

He was no neophyte, however. He knew the way of things with planning parables. He picked up the third wax tablet, and before turning his mind to the true challange, began to ponder the same parable with only 4 subjects. There would most likely be a pattern.

The girl filled his third cup as he worked. He barely noticed, engrossed in his efforts. Soon he had filled his wax tablet. He had smeared and re-used all the space he could, so he reached for the other two. His master said nothing as he spread his work across all three surfaces.

The second cup of tea began to do its work as well. His fingers ceased to shake, his movements growing precise even as they kept their nervous speed. The buzzing in his head faded. He was left with an immense clarity. He felt as though his mind was skating over vast frozen rivers of thought.

No matter what he tried, though, he was unable to make the parable yield. He could generalize the pattern from three, but was that the best he could do? Without enumerating them all, how was he to know?

The glade faded from his perception as his focus intensified. His eagerness to learn the nature of the universe was long forgotten at this point. All that remained was a desire to resolve the parable. His world shrunk to encompass only his hands, his eyes, and the three tablets.

What felt like mere moments later, his master’s voice dispersed his focus. “You may drink from the third cup.”

Everett blinked, startled rudely back into reality. The sun was high above him now, burning the back of his neck from above the treetops. His master had not moved, but his granddaughter was no longer with them. Everett hadn’t even noticed her leave.

“Master?” he asked. He had not yet solved the third parable.

“You have understood the shape of the parable, if not its solution. You are ready to learn the nature of the universe.”

Right. That was why he had come here today. Everett leaned forward and grasped the third cup. The leaves inside had dissolved, producing a green liquid so dark it was almost black. He tilted his head back and drank. The dark tea was cold as it slid down his throat.

The new liquid reacted violently with the contents of his stomach. Everett doubled over, the cup half-finished, clutching his abdomen as the contents began to bubble and roil. His hands clenched. Sweat began to roll down his neck, and then his forehead, stinging his eyes.

The violent bubbling feeling spread through his body, until even his fingertips felt like they were fizzing. His brain jumped around like the lid of an overboiled pot. The clarity from before was shattered, shards of his attention darting all over the glade, unable to form a coherent thought.

He was going to throw up. Or was he?

The world split in two.

In one world, Everett retched. Thick green liquid and chunks of his breakfast soaked the pristine grass. He fell to his side as his whole whole body relaxed, free of the poison. His mind quieted. His thoughts returning to a single track: failure.

In the other, the liquid stayed down. The strange feelings intensified. His mind grew more and more scattered, and his panicking body began to feel far away. He was going insane.

Everett felt a light touch on his forehead. The first world dissolved, and he was left in the second, except that his master had stood and was pressing the back of a hand to his brow.

“Focus,” his master said. “Consider the first parable.”

Everett complied, repeating the parable to himself.

Every year Lord Midas summons 2 of his subjects to court. They are taken to separate rooms, and each made to flip a coin. They are then asked to guess whether the other man flipped heads. If they are both right, they both go gome with a treasure. What should they do? And what are their chances?”

He saw the parable in his mind’s eye, the two men standing in separate rooms, each flipping a coin. They both flipped, and the world split into 4. Then, in each of the 4 worlds, they wrote down their guesses according to the strategy he had found.

Everett saw it. He saw it! He grabbed a tablet in the real world, he wasn’t sure which one. He smudged his existing work off with the back of his hand like a neophyte, and began to write.

BEST STRATEGY:
Flip H -> Guess H
Flip T -> Guess T

FLIPS GUESSES SUCCESS PAYOFF
H H   H H     1 1     1
H T   H T     0 0     0
T H   T H     0 0     0
T T   T T     1 1     1

The two men were always both right, or both wrong. If he compared it to the naive solution of both men always guessing heads…

FIXED STRATEGY

FLIPS GUESSES SUCCESS PAYOFF
H H   H H     1 1     1
H T   H H     1 0     0
T H   H H     0 1     0
T T   H H     0 0     0

That was the secret, then. Comprehension bloomed in his mind. It was true that since the two coinflips were unconnected, neither man could guess correctly more than 1/2 the time. If their guesses were unrelated, they would both be right 1/4 the time.

But they could force their guesses to be related by looking at their own coin. No rule of the universe said they couldn’t. The number of correct and incorrect guesses were the same, taken across all worlds – the number of 1s and 0s in the success column must always be equal – but they could be moved between worlds by different strategies. Since the two men only got paid if they were both right, lining the 1s up like pictures on a slot machine paid off.

Everett, excited, moved on to the second parable. This one had 8 worlds, enough to strain his mind, but he persisted. He repeated the parable to himself and began to write.

Every year Lord Midas summons 3 of his subjects to court. They are taken to separate rooms, while Lord Midas flips a coin for each of them. Each is told how the coin landed for the other 2, and then offered a chance to guess how their own coin landed. They may pass, if they wish, and decline to guess. If at least one guesses right, and none guess wrong, they all go home with a treasure. What should they do? And what are their chances?”

BEST STRATEGY
Others Flip H H -> Guess T
Others Flip T T -> Guess H
Anything Else -> Pass (P)

FLIPS  GUESSES SUCCESS PAYOFF
H H H  T T T   0 0 0   0
H H T  P P T   P P 1   1
H T H  P T P   P 1 P   1
H T T  H P P   1 P P   1
T H H  T P P   1 P P   1
T H T  P H P   P 1 P   1
T T H  P P H   P P 1   1
T T T  H H H   0 0 0   0

It was beautiful. Once again, no individual subject could be right more than half the time. The number of 1s and 0s in the success column were equal. But they could pass, and so they could arrange all the 0s in the same world, and make sure each other world had exactly one 1. Since they got paid off if at least one man was right and none were wrong, they could win in 6/8 worlds!

Everett was enlightened. It generalized so cleanly onto the case of 7 prisoners. He just had to figure out how to concentrate 7 wrong guesses into the bad worlds, while having 1 correct guess in each of the good worlds. If it worked, the payoff rate would be 7/8. With 3 prisoners, it had only been 3/4! Somehow adding more prisoners made the task easier!

Everett smiled, feeling the sun on his face, the wind on his hair. A moment later he vomited.

It was as worse than it had been in his imagined world. His stomach contracted, over and over, like the fist of god squeezing him dry. Black-green goop mixed with fragments of leaf coated the glade. He just barely managed to avoid getting any on his robes.

“Good,” his master said, his voice warm. He had retreated a step just before Everett retched. “You have comprehended the nature of the universe.” His master turned to leave, while Everett struggled to collect himself.

Everett pushed himself up into a more dignified position, coughing the last bits of liquid out of his throat. He had certainly understood something. If you thought of every random event as shattering the world into pieces, you could decide how to behave by simply counting how many resulting worlds had the property you wanted to achieve. But one question remained.

“Master,” he called to the old man’s retreating back. “I still have not comprehended the third parable.”

His master waved vaguely over his shoulder as he walked. “A lesson for another time. You lack the tools to truly understand. We will return to the final parable next year, when you have comprehended the first mystery of Hamming.”

And with that, he was gone, disappearing into the forest as easily as he had arrived.

Everett sat for a moment, collecting himself. His head felt stuffy, like someone had been cramming cotton balls up his nose. But he set his jaw. He wiped the last drops of vomit with his sleeve, and picked up a tablet.

The first mystery of Hamming could go stuff itself. Armed with the nature of the universe, he could figure it out on his own.