The Impossible Shuffle

If you're the type to have fun with numbers, you've perhaps encountered the outrageous number of possible shuffled decks using a 52 card deck. It's 52! and I'm not shouting. But in other notations, that's about 8.06 * 10^67, or 8065817517094387857166063685640376697528950544088327782400000000000. The popular claim is that there are more combinations than there are atoms in the known universe. But I know a lot more universe than most of the folks making that claim, and I'm not sure the math adds up. What's certain is that mortal, finite humans will never, ever, ever, see all of them naturally.

But that's not just a statistical certainty, or a psychical limitation. There's more to it than that. There are at least 5 combinations that are prohibited from forming, the universe itself won't allow it. The chances that we should ever discover any of them are impossibly small, and even if we did, would we even notice? Well, I've never let a small infinity get between me and anything interesting, so sure enough, I found one.

I was working on a battery powered automatic psuedo-shuffler when it happened. A device that appears to shuffle cards but actually arranges them carefully. It's subtle, when you try to assemble the cards in just this specific order with a shuffle, one will always slip out, or miss its spot, or otherwise refuse to settle into a nicely shuffled deck.

It's identical to a simple mistake which happens all the time, and which is quickly rectified by shuffling the cards into virtually any other order. That's some high class, custom, coded-to-the-metal magic if I've ever seen any. The kind that usually means some eternal being or another is involved.

The obvious place to start was Lady Luck. As the most prominent overseer of games of chance, fairly shuffled cards were a point of pride to her. For that very reason, she almost sent me packing with a curse until I bet her I could prove my claim. I carefully arranged the deck's precursor, split it, and meticulously shuffled card by card in a manner so tedious the Lady took to audibly yawning. But about 3/4 of the way through, a card fluttered out to the floor.

"Aha! See? The card refused to join the deck," I said.

"You dropped a card, and you're blaming the whole universe for it?" she asked, but she knew I wouldn't waste a whole trip just to fumble around. "Do it again."

I did, with the same tedious rigor, being especially dramatic with how carefully and securely I moved each card. Sure enough, a card slid off the pile anyways. I did it again, and one of the cards ended up flipped, something we both knew shouldn't be possible. I began to do it again and she stopped me.

"I've had my temple defiled enough for today. Thank you for bringing this to my attention. I trust you're not going to go spreading rumors about this?"

I shook my head. "Of course not. That's why I came to you first. Honestly, I figured this was something you were doing."

She scoffed. "I find fairness and chance to be beautiful components of this reality. This wasn't my doing, I wouldn't tolerate it. But that gives me a suspicion about who to ask."

I saw my opportunity. "Did you memorize the order yet, or do you need me to write it down?"

"No. You never quite finished your shuffles, remember? And the set-up was too deliberate to be my domain."

Perfect. "Then I guess you'll have to take me with you as you track this down."

Her response chilled my witty confidence. "Make no mistake, little wizard, you're not leaving my sight with that configuration still rattling around in your head. That's dangerous knowledge, in ways you can't even imagine. Luck must be pure, or a whole lot of things are going to come crashing down. You're coming with me, and we're going now."

"I'm always ready to travel," I said nervously, "but where are we going?"

"To see my cousin. The weird one. You call him Entropy."

Lady Luck's realm is dressed up like a casino lounge. Maybe that's just a show for when humans come around, but it fits her demeanor, and it's comfortable. Entropy's realm was utterly alien by comparison. Uncomfortably tight, endless hallways through some kind of mega-machine that produced grating, discordant hums and whistles. The air tensed up and pulled at itself in something that certainly wasn't wind, but I had no framework to understand. Underlying it all was a kind of mathematical precision that is typical of astral constructions, but at a scale far bigger than I could imagine.

I opened my mouth to ask about Entropy, but the Lady shushed me urgently. "Tread carefully, we are not part of their design. This means we are anathema, and they have surely already noticed. They're recalculating futures to account for this deviation to devise a proper model, which-"

"Which I use to optimize this encounter," said Entropy, appearing from around a corner. "Which, I will add, is utterly ruining decades of planning. So, as precisely and succinctly as you can, explain your intrusion. Immediately!" They didn't bother with a human appearance, instead I saw a polyhedral shape with eyes and mouths on each face. They spun between faces in a bizarre mockery of verbal speech, but it did the trick.

Lady Luck took the charge. "Someone prohibited possibility, specifically a possibility of cards. Was it you?"

"Preposterous, I have no interest in your silly cards. Is that all?" they replied with a matter-of-fact tone that I found pretty convincing.

"I'm sorry, cousin, but this is too big for you to ignore. There's a combination of cards that can't be shuffled. Real forces bend around it, I've seen it with my own eyes."

Entropy spun to lock a particularly stern set of eyes on me. "Abandon any hope of trickery, magician. If you have fooled my cousin you may be quite clever, but no angles are hidden to me."

I believed them. "No tricks, and I'm a wizard." The Lady shot me a warning glance. "I'll demonstrate plain and simple, look at it every way you'd like."

I pulled the deck from my pocket, and began to prepare the cards. This agitated Entropy, but Luck calmed them down with a few sentences in a language I'd never heard before. Once it was ready, I announced "The following shuffle isn't possible". I shuffled the cards together meticulously, and just like before one slipped out. "There it is. That always happens."

As I said the word "always", the realm lurched around me. It stretched in hyperbolic ways, churning through dimensions I usually prefer to keep in one place. It looped in on itself, and I saw myself performing the shuffle a million, billion, infinite times. In some the card fell, in some it flipped, in some it phased right through another card. Every possible way for the shuffle to fail, it did. Then this all snapped back to a normal space with a zapping sound I still don't like thinking about.

Entropy didn't waste any time. "Agreed, this reeks of tampering. It was not me, I would never. Neither would you," he said to the Lady.

"Exactly. This is sacred to me. I need to know who would dare."

Entropy thought for a long time, which was mere seconds, "Oh, of course. The Twins. Just like I said, at the Gravity Games. Remember?"

She frowned, "This isn't your idea of revenge, is it?"

"After all this time you still doubt my character? Begone, and tend to your own mess."

We returned to Luck's lounge, and I got to ask a few questions. Gravity is one of the eternal forces, but it's just too dumb to be interesting. Stuff moves together, that's it. Which means it doesn't really require a figurehead, just someone to turn it on in the beginning and off after the Seventh Squish. So before Time started counting, Mythos organized the Gravity Games to determine who's job it was to operate gravity.

It came down to a card game, between Luck, Entropy, Space, Time, and Physics (who was going by Matter at the non-time.) I don't know the game they played, I heard a name but it didn't have many familiar letters in it, and I doubt I could even comprehend the rules. But it's played with the familiar 52 card deck. That might surprise you but it's not really a coincidence.

Matter won after four rounds. But the game was overcast by an accusation of cheating. Lady Luck found the idea blasphemous, and insisted the cards were fairly shuffled, and no one could prove otherwise. If anyone could, it would undermine her role, and the universe requires far too much luck in order to turn out interesting. So the game was ruled fair, Matter won and took the prize, becoming Physics.

"Physics is a solid player", the Lady said, "but they stick to simple, straightforward strategies. I'd wager my favorite coin that they played an honest game. But why would anyone else cheat the game to favor Physics?"

"Did everyone want to win?" I asked, still unclear on weather gravity was a prize or a job.

"Of course! Gravity is the easiest thing there is. You could probably even do it, if it weren't for that pesky mortality. But it's everywhere, so the pay is really great. It's a dream job, and doesn't even get in the way of a regular one."

Pay? The Eternal Forces get paid? I had to fight of a swarm of questions swirling in my mind. "What can you tell me about the other two players, Space and Time?"

"The twins," she said. "They are trouble. Taken on their own, each is a reasonable person. But when they are together, they just bring out the worst in each other. It's the relativity. You're on a causal timeline, right? So you're in a good spot to understand that space is the thing that takes time to traverse, and time is the thing it takes space to traverse, right?"

"Yeah, they're all wound up with each other."

"Right, but that's because they are really just one force, spacetime. No one expected twins. So they are sharing the same force, and they have all these silly rules they've come up with to determine their domains. It's a sibling squabble played out across the cosmos."

I was still confused. "How does that card game connect to the prohibited shuffle?"

"I detest this thought, but it may have been done to protect me. Our game includes a significant bluffing factor. If someone were able to examine the state of the deck carefully, they could reverse engineer the state of each hand, and figure out which bluffs were real. This would allow someone to prove if any cards were cheated, which throws the fundamental fairness of chance out the window."

"If someone cheated just to undermine you, why would they protect you? Blackmail?"

"No, blackmail is too risky against the arbiter of luck," she said. "You're right. That doesn't make sense."

"Unless it wasn't about you. Someone wanted to control the result, but didn't want to threaten your domain. You've said Physics wouldn't cheat, but how certain are you of that?"

"About as certain as a sunrise. Physics and Entropy are rigid minds, their power comes from their predictability. The idea of behaving outside the rules is beyond their scope."

"That leaves the twins. You've said they are mischief together, could that include cheating the result for some reason?"

She sighed. "Perhaps. But I don't think they had the capacity. Tricks of space wouldn't fly with Entropy in the room, and Time wasn't fully empowered yet. Oh.. oh no. Those brats! They got outside help."

"From who?"

"I don't know, but we're going to find out."

I've met Father Time on a few occasions in my adventures. For a short bit we played Go by correspondence, his moves arriving as a chime on a nearby clock indicating a position encoded into the values of the hours, minutes, and seconds. I wasn't a strong enough player to entertain him for long, but in every interaction he'd always been an old sage that spoke simple truths kindly. The thought of him as a mischievous sibling was a bit amusing, and a bit hard to believe.

His realm was new to me though, it's not one that random wanderers like me will ever find on accident. I expected a clockwork world, so I was surprised to find a garden instead. Time flowed meticulously through the plants, allowing each to bend and sway naturally, but in time with some ethereal music that permeated the space. As I watched, I realized that many of the plants were actually slowly traveling in an orbit around the massive tree in the center of the garden. Father Time was there, napping under the tree.

"Wake up, old man!" Lady Luck walked up to loom over Time. "You have visitors."

Time rose wearily, stretching and yawning before finally opening his eyes. He saw me first, and wondered "What's a wizard doing here?" Then he turned to see Lady Luck, and in a flash of embarrassment he shifted to a well dressed appearance. A well fitting suit, a glamorous robe. His bedding disappearing into some unseen space. "My apologies, Mistress Fate. I wasn't expecting company."

"You know I haven't used that name in ages, Time. I'm here to talk about the Gravity Games."

"The Gravity Games?" asked the old man. "They were so long ago. Why now?"

"You can drop the old man act, this is too serious. I need to know if you wanted to win."

The old man furrowed his brows. "Didn't everyone?"

"I don't think so. In fact, I think someone wanted not to win so badly that they cheated to ensure it. I just don't know how."

Time sighed. "I'm not admitting to anything, OK, but just imagine what happens if I became the arbiter of Gravity. Sure, it's a simple force, but it's a big one. Back then, I was so weak, the flow of time felt so inconsequential.

It clicked in my head. "You didn't want to become Gravity. If you took over Gravity, Space might just take all of spacetime to balance the roles."

"I like what I do," Time said, "I like my sister, I like our arrangement. But I don't know if I cheated."

Lady Luck looked like she was ready to explode. "If you tampered with the game.."

Time shook his head. "I don't know, like I said. Maybe I did, maybe I didn't."

"Quantum" the Lady said, halfway between a question and a statement.

Time "It was just a conversation. We didn't shake hands or anything, I didn't give them anything. I don't have to tell you the dangers of compromising luck. If the little guy did something, he did it in a way we're not supposed to detect. Maybe he didn't do anything."

"Well, someone did something," I said, pulling the deck of cards from my pocket. I gave a simple demonstration of the impossible shuffle.

Lady Luck explained "There's no need to obscure this deck configuration unless it's to cover up tampering."

A small man appeared. Appeared isn't quite right. He went from not being there, to having been there the whole time. Quantum.

"I had hoped this wouldn't come to be, but I suppose I owe you an explanation."

Lady Luck stared at the man, her fury barely tempered by her tight lips. "It had better be a good one."

"I admit, there is a small number of impossible deck configurations. I'm sorry I didn't tell you, but it wouldn't make a difference if you knew. They include the four decks you shuffled that day, and a few more for obscurity."

"You gave cover to cheating. This is a violation of the highest order!"

"Did I?" Quantum asked. "If you have any proof of a cheating action, I'll take the blame for obscuring it. I admit only to hiding the deck configurations. Honestly, I expected Entropy to be the one to notice, with all their rigorous machines. That a mere human wizard caught it is such an act of improbable luck that I'd question if you were involved if I didn't know better."

I felt very, very small. Like a kid at the adults table for the first time. "For what it's worth, I don't have to tell anyone. I can keep this a secret." All of them turned to me, as if they were surprised I was still there. "It seems to me that whatever did or didn't happen, the best outcome is for no one to know for sure."

Quantum didn't seem to pay me any mind. "All that we are left with, in the end, is our beliefs of what happened. And for what it's worth, Lady Luck, I think you should believe in yourself. Your presence is important here, and especially to me."

The Lady took a deep breath. "I'm not sure what I believe. But it seems like the truth has been hidden from every angle. Should I ever find your fingers on the scales of fate, they will be removed."

Quantum smiled, "I'll hold you to that promise." Then he turned his gaze back to me. He and Luck exchanged a few words in that language I didn't recognize.

Then the Lady turned to me and said "You are a loose thread, wizard, but without you this wrinkle may have gone unnoticed. Entropy has already begun the process of editing your memories."

I'm no fan of entities messing around in my head, but I didn't feel like I had any grounds to object. "Let me keep the story."

Lady Luck's eyes turned curious. "What exactly are you requesting?"

"You know I have a storyteller's heart. My stories are my essence, and this one is particularly wild. Strip the truth from my memories as you must, but let me keep the story."

She considered for a moment, then chuckled. "Your stories are already fiction, I suppose it wouldn't hurt. Fine, you may tell the version of this story that you remember, if you'll abstain from hunting for the prohibited configurations."


"Before you go, I need to check on Entropy's work, for my own sanity. Tell me the first 5 cards of the prohibited configuration."

"Let's see, it was the 4 of Hearts, 7 of Diamonds, 6 of Shears, the 8 of Shears, then the 2 of Crosses."

"Perfect" she said, punctuating it with a hard snap back to my studio. And then she was gone. A short while later, I had to look up what a spade was.

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