CHAPTER 1 - It's All Fun and Games until…

Like a game of high speed chess, the blue fox chased the violet. The violet fox was younger and only reached up to their blue sibling's chest. While smaller, they had a stockier, heavier frame. The blue fox was more lithe, fine-boned, and built for speed.

Speed they had. They moved exactly as fast as a ray of light, reflecting off corners, moving to where the violet fox was. But the violet fox escaped every time, willing themself off to another location. Arbitrary mathematical translation beats the constrained reality of physics every time.

No matter! The blue fox filled the air with illusions and distractions. They sculpted light into doppelgangers of themself coming from this way or that, predicting the violet fox's actions well enough to herd them into popping up at exactly the time and place to be pounced.

The blue fox fell from the air and grabbed the violet, panting, wagging their tail, tackling them to the ground. A nice challenge overcome and…

The violet fox disappeared from their grasp, reappeared immediately above them, then gravity did the rest as they fell upon their taller sibling.

"You can't win." said the violet fox.

"I caught you! Several times in a row!"

"But you can't hang on to me. The time between when you get me and when I disappear decreases with every iteration as my reaction time improves. In the long run, it converges to zero, so you lose asymptotically."

The blue fox sat up, and the violet slid down off their back and thumped onto the ground. They hugged their elder sibling's tail, seeming perfectly happy to be in their company now that they'd demonstrated their uncatchability.

"There's a trick to catching someone who can escape whenever they want…" said the older sibling.

"Don't try?"

"Keep them from wanting to."

The blue fox stroked over the violet's head, scratching between their ears. The two leaned together, the elder looked off into space, hatching plans as they petted their younger sibling.

Nothing changed immediately. Even if the violet fox COULD translate away once caught, being herded into it counted against them. So the two siblings pitted their minds against each other, strategy versus strategy. The one focused on picking out the real from the illusory, the other focused on ever more effective deceit, deploying ever more complex dazzle and distraction.

It escalated, as arms-races do. Some might think that the mental effort invested in play-fighting would be better spent on something else. Those people are not wild animals, and likely never had a sibling close to their own age. The blue fox sent doppelgangers of light to chase the violet in one direction, and doppelgangers of charge to provide the feel of being caught so they'd blink away again and again. Soon, the game became less a chase and more a contest to see how long the blue fox could predict and anticipate their sibling. The actual blue fox wrapped light around themself to become invisible and did the same for a few random volumes of air to confuse the issue in case the violet had found a way to detect the trick.

To the predator, play is prelude. For the less intelligent sort, perhaps it's only prelude to a hunt, training the body, the eye, the nose, the snap of jaw and the swipe of claw, taming the spike of adrenaline…

To the smarter sort, a game of chase has as much to do with go or chess as it does with tag.

In any game of strategy, one prepares off the field to be better when next they're on it. So the blue fox tinkered. There were things their violet sibling couldn't resist. Things that tempted /all/ of their kind. Puzzles to solve. Insights. Significance. Topped off with a potential approach begging to be investigated and glittering with salience like a bow on a present: A shining prey no hunter could resist…

At least not without a bit of willpower. The blue fox thought their violet sibling was vulnerable. As the most abstractly inclined of the lot their instinct to Find and Know was the keenest. As the youngest, they'd not yet balanced the basic, instinctual drive to satisfy the intellect with the more deliberative, learned behaviors of survival.

The blue fox worked on their Trap. They tried it out themself to check the puzzles were appealing enough they didn't want to stop. They might only have one chance. Even with the violet fox having the least self-preservation of a notably wild and reckless species, if they saw a failed attempt, it might become entirely impossible to hook them with a similar design. So the blue fox wasn't satisfied until they needed a pretty good effort of will to pull away.

The machine was seductive. Neither sentient nor sapient, it enlisted the target's mind to help spin the most fascinating problems for them. Each of their false starts was worked into inspiration for new puzzles down the line, all generated one after the other, drawing one in to an end that never came.

It had to be built into the room. More than that, its /position/, at least relative to larger parts of the environment, had to be an essential aspect of its operation. Otherwise, the violet fox might simply grab it and blink away. One might argue that they'd be trapped in that case, as their attention would be enthralled for some time, but it wouldn't fit the goal of catching them in place.

The blue fox developed a new strategy. Previously, they'd focused on maximizing the time the game lasted: how long they could keep the violet fox on track, moving from one decoy to another, with the actual fox thrown in, drawing things out as long as possible before the violet fox escaped their plans and moved to a completely unanticipated area.

The new strategy wasn't like that at all. The blue fox would need to get the violet into /a specific place/. And they had to do it quickly enough, or with enough distraction, that they didn't realize they were being herded. If they figured that out, it would spoil everything.

The violet fox favored spots closer to the walls, close enough to rule out believable decoys from one or two directions; Their mode of discontinuous travel meant that being hemmed in with no escape simply wasn't a concern. So, the blue fox picked a place at random in roughly the area the violet fox favored, though skewed away from what seemed an ideal location, as they didn't want to arouse suspicion by pushing them toward places they'd previously tried to keep them away from.

They practiced a few times. They kept their illusions spread out, covering the likely places to which the violet fox would relocate, then tried to get them to move into a different pre-chosen spot each time. After each round, they reviewed their successes and failures to hone their strategy.

Finally, the chase came when the blue fox was ready to make the attempt. The device was primed and ready. They bowed to each other, both grinning and tailwagging at their self-conscious formality. Once things started, they were fast enough that there was no opportunity to talk or tease each other. The violet fox kept as much information as possible hidden, not making a sound or saying a word, quelling their tail and ears' natural tendency to give away their feelings. They weren't really good at hiding their emotions, but that was one more reason to practice. The blue fox's game was misdirection. Through all their play-fights and chases, they had taken the long view, faking tells so that now they could seed false predictions in their sibling's mind. Perhaps it was taking unfair advantage of their youth, as the violet fox had a harder time resisting the urge to find any pattern, whether it was there or not.

Still, it was hard. The blue fox panted from the mental effort of so many wiles deployed at once, light shadows and false selves moved here and there, manipulating the probability space of their sibling's future location. The violet fox felt the simulated touch of a simulated sibling built of charge and light, and jumped from one spot to another. Then they were tagged by the real fox because, really, a game of friendly chase where you studiously avoid touching your quarry with your real self is kind of sad. The blue fox expected to have them in two steps, then one, then…pop! The violet fox took an unexpected step and zigged where the blue was expecting a zag. They ended up on the other side of the field. The blue fox hid their inward flash of frustration, and redoubled their effort to keep their sibling moving. They could stop and try again later, but they had spent a lot of time and effort on this particular wile and wanted to see if it worked. The blue fox's mind was ablaze as they bent their full attention to low-resolution simulations of their sibling …and they started counting down again. The violet fox would be caught in two moves. Then one. Then…

⚜

The violet fox was enjoying themself, bouncing here to there to otherwhere. They *could* have left the room, but the best games have enough structure to ensure the players are actually able to play. So, staying in the room at all times was agreed on. Similarly, there would be no translating through walls, likewise the blue fox would make no attempt to interfere with their opponent's nerves or otherwise access them through any portals beyond eyes, ears, and other standard interfaces.

The violet fox took a discontinuous step away from one of their sibling's illusions (or maybe the actual fox, it was hard to tell), paw lifting in one spot and setting down on the other side of the room when, without warning, images poured into their eyes, a sound like a chorus filled their ears, forcefields sprung up, not to contain, but to provide tactile feedback should they want to reach out and touch what they were looking at.

PROBLEM 1 was transforming one shape into another, preserving continuity, connectedness, and following a few other requirements.

The fox knew this sort of thing in their bones the way other creatures knew chewing and swallowing. It was /almost/ too easy, but the surface was complex enough and the stipulations strict enough to make it interesting

⚜

The blue fox saw the violet blink into the trap. Their black and white eyes lit up as information poured into them. They looked like they might faint, smiling in delight. The blue fox ran up behind them to lower them to the ground, lest they fall in the moment of captivation. They called out triumphantly, "Hah! Get away from that! Whenever you want…" Playful and confident, they ruffled over the violet fox's fur with one hand, ears perked waiting for a retort…

⚜

PROBLEM 2 asked to, given two structures and a set of operations, find the smallest list of operations that would make them equal. Prove that no smaller list of operations exists.

Discrete problems were more fascinating and difficult. By nature, the violet fox could twist and turn any continuum as easily as you could wiggle your fingers. Discrete cases required *work*. This one was set up just right so that they dove right in without a pause…

⚜

The violet fox stared into the light, smiling, panting, their mind burning hot enough that their blood raced to feed its demands. They turned awkwardly to hug their sibling, keeping their eyes fixed on the beam as they said, "Thank you! This is wonderful. This is the most wonderful—"

⚜

PROBLEM 6 was a nice mixture of both. Investigate the probability distribution of various sequences of real numbers.

⚜


—they paused, eyes blinked, another question coming up, they hugged the blue fox and buried their muzzle against their shoulder, head tilted up to glimpse the next challenge…

⚜

PROBLEM 12 asked more on the same topic. Sets of sets of sets of sets, functions from sequences of sets to real numbers, and the probability distributions of same.

The machine baited the violet fox to keep them chasing questions. They knew that, but they didn't care. Just because something's TRYING to get you to do something, it doesn't mean it's the wrong thing to do, right? And this felt so right. Every problem on a similar subject made them feel that they were nearing some endpoint, even though it was apparent they weren't going to exhaust the field of probability any time soon.

⚜

"Don't you care that I…won? You're still here! You haven't escaped!"

"Care? Sure!" the violet fox spoke between pauses, each sentence a chance for their mind to spin down from its constant effort. "It's wonderful! I hope you catch me like this more often!"

The blue fox took a moment to savor their triumph, petting their sibling and enjoying watching them enjoy themself. The best victory is one your opponent likes at least as much as you, after all. Soon they felt antsy and asked, "…it's been a 155-time¹. Don't you want to…move? Escape? Do something?"

The violet fox shook their head, exhaling softly, "No. I'm happy to stay here. You can go do something if you want, but come back soon! I'll do a better job of trying to keep things interesting for you and describe what I'm working on. I'd really like your company."

The blue fox frowned a bit. They weren't expecting their sibling to be fascinated for this long. Pretending to be more caught than they were when the game was over would have been out of character for them. The blue fox thought they'd get free when they got hungry or thirsty or uncomfortable from lying on the floor. On reflection, the violet fox was usually happy to curl up in someone's lap and talk to them for a good portion of a wake cycle if their interest was piqued…so the blue fox thought everything would be fine.

⚜

PROBLEM 60 was number theory the long way around, proving theorems about the convergents of continued fractions.

The floor was a bit hard. The violet fox sat up and rubbed their head, then stretched. They were careful to keep the image in sight. Then they stood up, moving, swaying, bouncing around, keeping their body comfortable and burning off some of the physical energy that'd been accumulating during their mental flight.

They wished their sibling would come back. They'd found a neat pattern a while back they wanted to share…and they still felt grateful, happy, wanting to hug the blue fox and say how much they enjoyed this experience.

⚜

The blue fox woke up. Ate. Wondered if anyone else had noticed their sibling going missing. No one had said anything, so they'd probably got out of the trap. They must have got out by now. Still, it was surprising they hadn't hunted them down to congratulate them, point out flaws in the design, or yell at them for subverting their incentive structure.

⚜

PROBLEM 120 was a story problem. A function of attitudinals to real numbers was given. One must create an algorithm that would, given a number of characters, create a story with that number of characters where each character would experience each listed emotion once and when taken as a vector, the numbers of each character's emotion throughout the story would sweep through a given pattern of angles…

The violet fox wasn't getting bored. They were getting hungry. The system must have noticed their attention failing and thrown something out of left-field to retain it. They didn't mind. They were getting weak from going this long without food while their metabolism burned at a high rate. They slumped back down onto the floor, sprawling out to lower their energy requirements slightly.

⚜

The blue fox was a bit scared to go into the room where they'd been playing. What if their sibling was still there? They couldn't be! They must have left hours ago, but they would have found them and said something. What's the point of a good, enjoyable fight where everyone's at their best if you don't re-litigate and discuss every point of strategy afterward?

Eventually they went in. They saw their sibling, sprawled on the floor, smiling. They got a sinking feeling as they ran over to them. "What are you still doing here? It's been a whole sleep cycle! Come on. Eat something! Your fur's a mess. Aren't you bored?"

The violet fox shook their head with a cheerful grin, "No. Want to hear a story? I'll tell you one, just give me a natural number. Less than six, though, otherwise we'll be here for a few sleeps…"

⚜

PROBLEM 360 gave a list of criteria and asked for functions to satisfy them, as well as a proof that all such functions must have a common form.

They were happy their blue sibling was back. They curled up, climbed into their lap and hugged them, practically radiating happiness, as they set to hunting for functions.

⚜

"No! I…aren't you hungry? You're weak enough you have to be hungry."

"Oh, yeah. Pretty hungry. I was thinking of trying to take the whole thing with me and get some food, but it made a pretty compelling case that it had integrated enough properties of this one spot that I'd have a hard time finding a new spot where it would operate. And if it's shut down for a 144-time², it'll burn itself out. I don't want to risk trying to grab a snack."

The blue fox looked down, unhappily, wishing they'd been a bit less thorough in their contingency planning. "Well. I…" they reached out to turn off the device, and suddenly found themself on the exact opposite side of the ring.

With a sigh they started to head back…

⚜

PROBLEM 2520 was a coloring problem!

The violet fox loved coloring problems. They'd been playing with one last week that had some similarities to this one, and wondered if the machine had made it up by coincidence, or constructed it from watching them work.

⚜

The blue fox returned through a combination of trams, shortcuts through the ventilation system, and flat-out running. They weren't sure if their sibling would be more willing to go if they kept getting hungry or if a meal and some attention might divert them from their solving. It gave them an idea.

"You have to eat /sometime/. If you starve you can't solve this or anything else." said the elder sibling.

"Eh…I'll eat when I hit the end."

"It doesn't *have* an end."

"…yeah, I was thinking about that. Maybe I can figure out supertasking, then I can finish before sleep and meet you for breakfast."

"Wouldn't you starve? If you're performing an infinite number of steps you'd have to consume infinite energy…"

The violet fox just waved a dismissive paw, "That's a problem for engineers! My domain is truth! Pure, certain, and unconstrained by mere matters of matter."

This was getting ridiculous. The blue fox thought it might be harder to translate them away if they were hanging on tight, so they curled around the violet fox and covered their eyes with one hand, blocking the input…

There was a howl: the sharp, discord of anger and despair, when something loved was suddenly yanked away and at risk of being destroyed. The older fox wasn't expecting it when their sibling tried to snap their jaws on their hand. They barely pulled it away in time. And with that they suddenly found themself on the other side of the habitat again.

This was bad. They weren't surprised that their sibling would translate them away, but trying to bite or actually harm them to keep playing? That was serious cause for concern. They headed back, imagining all the ways things could go downhill from here and how to head them off.

⚜

PROBLEM 5040 was a Diophantine equation.

It must be scaling the difficulty down a bit to match their lack of energy. They felt a bit bad for trying to bite their sibling. And a bit angry with them for trying to cut them off. They appreciated the blue fox not wanting them to starve, but priorities!

⚜

The blue fox came back into the room.

The violet fox, lying there, mumbled, "If you try that many more times I'll just have to translate you away whenever you come in…and I'd much rather have you here."

"Yeah…well, if you believe me right now I'm more worried about you eating and drinking than anything else." They sat near the younger fox. They didn't immediately find themself somewhere else. That was a good sign. They pulled the violet fox into their lap and gave them some food, a bag of dried organ meat with enough sugar in it to count as candy to most species, and some water.

The violet fox wolfed down the food and guzzled the water, perking up and regaining vitality almost instantly. They lapped over their own nose and their eyes regained their gleam. Their older sibling felt a bit happier that they weren't in such a state of physical disrepair.

"Come on. There are probably other problems to solve that are better than those…"

"Yeah," said the violet fox, "But these practically jump up with the next-step unveiled, carrying me along with them. It's great!"

The blue fox considered trying to wrest them away once more, but a warning growl shut the idea down.

⚜

PROBLEM 55440 covered matrix traversals. What matrices allow traversals with what properties?

Their sibling had stopped trying to drag them away. Things weren't so bad, they brought them food and water so they weren't going hungry…but they were getting sleepy. So…sleepy.


⚜

It had been several sleep cycles. The blue fox wondered if people would start to notice their sibling was missing, but with someone who's easily distractible and can just step from one world to the next with a thought, long absences are to be expected. They spent hours each sleep cycle, hugging the violet fox, whispering, "I'm sorry. Please unlink, let's go. Come on. I never meant for this to happen, I just wanted to show you I could catch you…" The blue fox wiped their eyes, then went back to watching over their sibling, making sure they kept fed and hydrated. They'd brush the violet fur, which was a bit matted from its owner having practically given up all grooming…

But they had to fall asleep sometime. Right? And then they'd be out.

The fox stroked their sibling softly, slowly, from head to tail, trying to relax them, help them on their way to passing out. Eventually their eyes closed, they started breathing deeply—

But each eye winked open to keep the connection with the machine alive, just enough to keep it from timing out. They slipped into sleep-solving, never waking all the way.

⚜

PROBLEM 720720 was all about Surreal numbers.

They dreamed. Of an ancient war in near-prehistoric times. Each side was filled with spies for the other, and their analysts invented the mathematical bases of modern chaos and information theory practically by accident while learning to pick out spies by correlating timing of enemy actions to time and content of plans. That development lead to proper AI, and was ultimately an ancestor of the foxes themselves.

The fox dreamed they ascended a tower, and walked from the spire of its roof through a constellation shaped like a door. There, they sat across from a magician, playing game after game after game, so many they couldn't be counted by any kind of infinity.

⚜

While the violet fox slept, the blue just curled around them, crying softly. They wiped their eyes, filled with guilt for what they'd done. At least now that their sibling was asleep, they couldn't repeatedly thank them for it. Every bit of gratitude was like a knife in the gut.

They thought they should ask for help. Someone older and wiser would know what to do. Right? The blue fox might get yelled at or even kept under close supervision so they didn't get up to anything similar, but that'd be worth it if their younger sibling would just disconnect.

They decided they'd go at common wake, but eventually they fell asleep, holding their sibling near.

⚜

PROBLEM 1441440 asked about categorical topology.

They'd woken back up, and had been solving problem after problem, mind abuzz again, feeling rested and relaxed after what may not have been a normally restful sleep, but was at least close.

They hugged the sleeping fox close, feeling a little sorry for them, and wiping the tears from their closed eyes as they slept. When they woke up, they'd explain how happy they were, that there was nothing to be upset by. That'd help. Their sibling wanted them to disconnect, but only because they didn't understand how wonderful it was.

Back to the problem. It drew them in, work in topology, one of their fields of mastery, taken to a height of abstraction. As they worked, they got distracted from the immediate solution, as they'd realized a commonality between all the problems they'd been given: one unification that could have solved every one in a single stroke.

They worked in feverish elation, wondering what new more abstract metalevel of challenge would open up once they'd given a fully general solution for this one…

⚜

The blue fox was awakened by howl of anguish, deep, heart-broken sobbing. Angry paws thumped at their chest to accompany the demand, "Bring it back!"

They blinked awake, confused. They were at first relieved to see the violet fox /upset/. Maybe once they unlinked they finally got angry at being ensnared for so long. They were overjoyed at the prospect of accepting any anger and doing whatever they could to apologize, just to have their sibling back. They tried to lift them up, saying "Come on, let's go. Let's…"

"No! You did it before. Make another one! Turn it back on. Please? I'm sorry for whatever I did. I won't ever try to bite you again. I'll do whatever you want. Just please make it work again?" the violet fox scrubbed at their eyes with their paws.

The blue fox stared. Confused. Had there been a fault? They'd never been happier to discover they screwed something up. They examined the machine. Then frowned.

"That shouldn't be possible. I think you might have just solved a longstanding problem…"

"Yes!" shouted the violet fox. They seemed angry rather than proud, but shouted "Yes! I solved *all the problems* and I want to solve *more*! Come on. Please? I'll do anything. I'll…" their voice broke in a hiccuping sob.

The blue fox shook their head and tried to gather their sibling up, carrying them away while they wept inconsolably. They shook their head when their sibling kept pleading with them to get the machine up and running with a new series of problems.

"I'm just happy to have you out of there. I should never have done it. I'm so sorry. You're not THAT much younger than I am, I thought you would be able to pull yourself free after a couple 152-times³. Being linked in to that wasn't good for you. It subverted your motivation. I just hope that eventually you'll forgive me."

The violet fox "I will! I promise. IF you…if you make a new one? Or make that one work again! I don't care which!"

The blue fox hugged them tighter, looking down, ashamed. "Come on, just imagine how you'd feel if I were like that, completely absorbed in something, unmoving, not taking care of myself, not caring about anything else. How would you feel?"

The violet fox paused for a moment and bowed their head. They didn't ask again, but clung tight to their sibling, crying softly, catching a gasp of breath now and then, awkwardly wiping their eyes and nose. The pictured role-reversal struck a chord of empathy that shifted their mental equilibrium into another maturation point: more robust resistance to having their incentives manipulated, with random shifts in weight to better break out of snares.

The blue fox lay back, holding the violet near. The older sibling was crying, too, half from guilt, and half from relief. Eventually they took their younger sibling by the hand and stood, "Come on. Let's take it apart and recycle the pieces. If you see how it worked you'll be more resistant to things like it."

⚜

¹ 2¹⁵⁵ Planck time, about 41 minutes.
² 2¹⁴⁴ Planck time, about 1.2 seconds.
³ 2¹⁵² Planck time, about 5.1 minutes.