The cell is spotless, as always.
I watch the fluorescent light flicker on as the cloying, synthetic voice makes its usual morning announcement over the loudspeaker. I ignore it – anyone would know it by heart, by now. It’s all lies anyway.
I roll out of bed and zip on my orange jumpsuit. 84 seconds later my teeth are brushed, my face is washed, and my hair is twisted in something passing for a bun. The wizards will be here soon, and I should maintain what dignity I can.
Most days, they make us wear hats. The hats come in many colors: most commonly black, followed by red, then blue. I keep track of the probabilities, though they don’t help. Once, when we were strategizing, Paul told me that he’d seen his own hat twice since he came here, during some of the wizards’ odder trials. I have never seen my own hat.
Sometimes, instead, we get light switches, or playing cards. One time they even gave us all contact lenses and left us on a desert island. None of us know why the wizards do this to us, but we play along. I imagine it would be fun, if we weren’t imprisoned and the stakes weren’t so high.
I hear a knock on the door, and stride over to open it. Each day they send a different wizard. Today it’s a blonde woman in dark glasses and a pantsuit, who gestures with the usual pair of handcuffs.
“I don’t suppose you say anything either, huh?”, I quip.
I can’t see her eyes through the glasses, but I can tell she’s staring at me.
“Well, it was worth a shot.”
—
Handcuffed, I follow the wizard into a room where 71 other people are already milling about. This suggests to me that we’ll be a group of 100 today, but I’m not certain. A couple of other prisoners look familiar. None are the sort I’d want to chat with.
Thankfully, as our number grows into the 90s, Paul walks in, guarded by a tall, dark-skinned man.
“Thank you sir,” Paul says, bowing his head to the wizard with an incongruous respect. “Hello everyone, my brain is open!”
A few of us turn and smile at him. Paul confuses me. He’s the friendliest of us all, even more than Emmy and Richard. He’s comforting – but out of place, perhaps.
The puzzle wizards leave us our written instructions. Each of us is handed a paper slip with a number from 1 to 100. I’m number 4.
In the next room, we are told, there are 100 matching number slips sorted randomly into 100 numbered boxes, with one slip per box. We each enter one by one, and have 50 chances to find our number. If we find it, we get taken back to our cell. If anyone fails to find their number, we all get executed.
“Well,” starts a bearded man in a tunic whom I don’t recognize, “to state the obvious, the naive solution gives us a success rate of 1 over 2 to the 100th power.”
“This is a challenge of probabilities,” says a man in the corner, his mouth still half full from the apple he’s crunching. “It’s like the blind card guessing games. We need to make sure we either all fail together, or we all succeed.” The man swallows before continuing. “Everyone think for at least 3 minutes, then speak when you have an idea.”
We sit in silence. Newton is controlling, but he’s clever enough that it’s worth following his commands.
3 minutes have barely passed, when another prisoner speaks up. “I have a proposal,” he says softly. His nearly-blind eyes point slightly above our heads. The man continues: “Everyone first opens the box labeled with his own number. Read the number contained in that first box, then take that as your next box to open. Repeat until you’ve won or run out of guesses. So long as there is no cycle of more than 50 boxes, we all win, otherwise we fail.”
“Wait. We might fail?” I interject.
The entire room looks at me.
“Of course we might fail,” says Newton. “There’s no way that any individual has over a 50% chance of success. Obviously, our total odds can’t be any higher than that. Euler’s plan gives us over a 30% chance of victory. That’s optimal.”
“Is it?” I respond. “We’re supposed to be perfectly logical mathematicians. Surely there’s some hidden information here that we’re missing that guarantees a win.”
Newton sneers. “There’s never any hidden information. We’re going with Euler’s plan.”
“But what if we lose? What if they execute us?” I ask.
“Are you in the first fifty?” Newton demands.
“I’m fourth,” I answer.
“Then enter the room when you are called and do as I say. Your questions are wasting all our time.”
I look around. A few people seem uncomfortable, but no one wants to contradict Newton.
Newton presses the call button, and the exercise starts. One by one, we’re called into the room. I try to hide my trembling hand and calm my heartbeat. After what feels like an eternity, the computer voice shouts my number. “PRISONER 4, ENTER!”. I open the door.
—
The steel door thuds behind me. I see 100 brass-colored boxes before me. I take a deep breath, and begin my search. I pull open box number 4, and see a paper slip numbered “61”. With a sigh, I turn to open box 61. Then box 10.
I’m 17 boxes in, muttering about how big a jerk Newton is, when I pause. That paper I saw at the beginning: did I read it upside down? Was it a 19 instead of a 61?
I could look again. But that might count towards my 50 boxes. Crap.
I’m not supposed to mess up. I’m a perfectly logical mathematician. I have to continue. I open another box. And another. And another.
As I slowly run out of boxes, my heart threatens to beat out of my chest. I gulp as I open the 50th box. The slip has a 24. No dice.
I hear the computer again. “Prisoners have failed. Begin execution.”
Before I fully process these words, a panel opens from the ceiling, revealing a terrifying mess of Tesla coils. I hear an electronic whine, then everything is light and pain.
—
The cell is spotless, as always.
The morning announcement accosts my ears. “Find the solution and you’ll be set free.” Fat chance.
I’d just been struck by some kind of lightning machine. And yet here I am, back in the cell. Could it have been a dream? I haven’t dreamed since I got here, but that’s no guarantee.
I walk over to the sink and splash water on my face. My working hypothesis is wizard shenanigans. It all seemed far too real.
Right as I finish zipping my jumpsuit, there’s a knock at the door. I see the same blonde wizard with the dark glasses.
“You again,” I snark at her, “Something else unprecedented.”
I study the wizard’s visage. Every hair is in the same place as it was when I last saw her.
“Is this my punishment then? For failing? I keep living out the day of my failure until eternity?”
She says nothing, but gestures to the handcuffs.
I let her handcuff me again. A year ago, I would have tried to punch her in the nose, but I’d quickly learned of the wizards’ power. For my punch to land, my fist would have to go half the distance to her nose, then one quarter, then one eighth, and so on for infinitely many steps. I could never make it – their magic knows no limits.
—
All 100 of us are back in the same planning room. Newton glares briefly in my direction. This time, I’m number 75. I catch a glimpse of Newton’s paper: he’s number 15. At least I’ll have some time in the room without him.
As I wait, I get a tap on the shoulder. It’s Paul. “So you were the last one in the room yesterday?” he asks.
“I was. I failed and got attacked by a lightning machine.”
“Oh, I know that one.” Paul says. “One of the more painful ones.”
“You know that one? You mean you’ve been executed before?”
“Of course!” says Paul. “It’s rare, but it happens. You wind up at the back of the hat line, or the odds come up poorly. You have to do the puzzle again, of course. And it hurts for sure.”
“Newton’s going to kill me,” I respond. “I messed up.”
Paul chuckles. “Newton needs to explain his findings better. He’s a clever one, but he’s rather rude. And it’s fine, there was a chance we were going to lose anyway.”
“No Paul,” I say. “I think I misread one of the numbers. We might still have won if I hadn’t made that mistake.”
“Are you certain that you misread a number?”
“No,” I reply.
“Then assume you didn’t. The puzzle wizards want to outsmart you, not trick you.”
“The puzzle wizards always trick us, Paul,” I say with a sigh. “Sure, maybe not in the puzzles. But every single morning, they announce that they’ll set us free if we find the solution. For 391 days I solve their puzzles, only to wake up in the same bed with the same voice and a new brainteaser to torture me. You can’t actually be dumb enough trust the puzzle wizards, can you?”
Paul falls silent, his brow furrowed. I can tell he’s trying to hide it, but his face betrays a sadness I’ve never seen. We stay in the silence for a few moments.
“Paul, I’m… I’m sorry. I didn’t mean to suggest–”
“You really think you’ve found a solution?”
Paul’s interruption catches me off guard. I recover after half a second. “Paul, I’m sorry for what I said, but you do have to admit that we solve the puzzles every day. I say the color of my hat and they show us the mirror and it always matches perfectly.”
“And you still think the wizards’ puzzles are what we need to solve.” Paul says, his sadness tempering into resignation. “I thought you’d know by now, but I suppose I wasn’t certain. I guess you don’t.”
Paul turns towards me in his chair. I can see the wrinkles around his grey eyes through his thick rimmed glasses.
“The puzzles will always be here,” Paul continues. “I’ve been captured by the wizards several times, since I was a little epsilon. Once when they wouldn’t let me return to America. Once when I was staying with John Conway. The wizards are a part of life. You can do their puzzles for years, and nothing will change. Not in this place. But when you see how far you’ve come, and how much you’ve learned? When you accept that you can be a great mathematician, even one of the best in history, without being perfect? Then, and only then, you’ll set yourself free.”
Paul holds my gaze for a second, then turns back to sit more comfortably. I watch him close his eyes and breathe calmly.
It’s another 7 seconds before I’m ready to speak. “But how can I learn?”, I ask. “I think my mind just doesn’t work that way. It just wants to torture me with this nightmare.”
“I cannot guarantee that’s false,” Paul says, keeping his eyes closed. “But I hope I’ve come to know you well enough to say that I don’t believe what you just said one bit. Now, shall we continue on with our exercise? I hear there’s a 30 percent chance we won’t even get executed again.”
I laugh. I’m not sure I fully believe Paul yet, but he’s never lied to me before.
—
We mill about the room. No one seems to want to be the one to press the call button. Newton is standing in the corner, tossing and catching another apple. I’m not sure where he keeps getting them.
A thought occurs to me. I call out to the room: “Does anyone else want to go over Euler’s solution?”
The conversation dies down. I have the room’s attention.
Newton speaks first. “Why? We already know Euler’s solution is optimal.”
“Maybe we do,” I answer. “But maybe seeing Euler’s logic can help us with a future puzzle, if Euler’s not here.”
“I can explain,” Euler says. “Each of us has a fifty-fifty chance of finding our slip. There’s nothing we can do to change that. What we can do is try to correlate our successes.”
“Got it.”, I reply. “It’s like the difference between flipping 100 coins that all need to come up heads, versus taping the coins together so they’re either all heads or all tails. Our total odds are better if we all succeed or fail together.”
Euler continues. “Yes, exactly. Now suppose we follow my solution. Prisoner number 1 finds an 18 in box 1, a 73 in box 18, and his own number 1 in box 73. He succeeds after searching through a chain of 3 boxes. But now we also know that prisoner 18 and prisoner 73 will see this same chain, and find their numbers after looking through 3 boxes as well. So long as there is no such chain of more than 50 boxes, everyone will succeed.”
I nod. “So the odds of us failing collectively are the odds of there being a chain containing more than half the boxes.”
“Yes. Which is the natural log of 2, in the limit,” Euler says. “Though I’ll leave that as an exercise.”
—
The wizards continue to lead prisoners into the room. A cheer erupts when the 51st prisoner is led in. If there was a chain of more than 50 boxes, someone would have failed by now. We’ve made it.
I find Paul. He’s chatting with some other prisoners in the corner.
“So Paul,” I interrupt. “Do you think this is it? Can I go home now?”
“Who am I to say?” Paul replies. “Maybe today, maybe a month from now. But I have a feeling that soon you’ll wake up back in your own home. And until then, let’s keep showing these puzzle wizards what we’re made of.”
I nod. “This was a strange one. But at least it wasn’t hats.” Paul smiles.
“Yes, my friend, at least it wasn’t hats.”
Coming soon: A deep dive beneath the Earth
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