Synopsis
Khazad dwarves are perfectly comfortable deep inside the rock, but their distant Luchiurp brethren prefer their above-ground workshops - especially when they're stuck in a wet, mouldy, bug-infested, Balseraph dungeon! Prisoners of a war they never asked for, Captain Mathos and company are stuck waiting for something - anything - to happen...
-----------------------
Notes
Hullo. This story doesn't track any particular game; it just takes place in Kael's Fall from Heaven universe. I'm assuming this is the right place to put it..?
If you are familiar with the mod, many characters and events will be familiar to you, but I *think* there's enough context in the story for it to stand on its own. If anyone gets confused on any given point and thinks the story should flesh out any particular detail, let me know and I'll see if I can tweak things.
Also, if you are more familiar with the universe than I am, let me know if any details just aren't right - I do want this story to be perfectly consistent with Kael's.
And of course alert me to any typos. I speak English well, but I don't type so good.
I've split it up into multiple spoiler tags just to improve readability a little bit. Let me know if you like it!
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The Circus Game
So what do we do? asked Mimbi, Do we all just say black and guarantee that half of us get out?
Guarantee? snorted Gambor derisively. I suppose youre sure the Clown King will put three black hats and two white hats down? The man is mad! We cant be sure of anything!
Hmm said Turi.
Oh thought Mimbi. I guess I just assumed theyd be randomly distributed. You know, on average, half would be black and half would be white.
Boffee pointed out, With an even distribution. And even if that is the case, there arent enough of us to rely on that. It isnt beyond the realm of possibility for all five hats to be randomly assigned the same color. Then he frowned again. Blimey, I cant remember my Posi Distributions. Being in the Companys rotted my brains.
Mathos scowled at him, Thats not even a Posi Distribution problem. Theres a one in thirty-two chance that all the hats would be black. Theres a lets see ten in thirty-two chance that three of the hats are black.
Boffee grinned, chagrined, but Gambor picked up the conversation, And a ten in thirty-two chance that only two are black. But none of that matters, because we dont have any reason to think therell be an even, random distribution!
Mimbi countered, But we dont have the slightest idea what else it could be, either, so we might as well just assume it is.
So we all just say black, then? Boffee asked. Would it be better for each of us to just randomly decide on our own whether to say black or white?
Hmm said Turi.
Mimbi frowned. That didnt seem right. Well we know that (assuming an even random distribution) the highest probability is that there are two or three black hats. So wed definitely want to make sure at least two or three of us say black. And since we dont know which of us might be black, wouldnt it be best for all of us to say it? Itd be a little embarrassing if everyone with a black hat accidentally said white, and everyone with a white hat accidentally said black, dont ya think?
Mathos sighed, That sounds like it could use a mathematical proof to back it up. Turi, youre the expert here. I dont suppose you could tell us which strategy gives the best chance of getting the most of us out of here?
Turi breathed in audibly, then let it out anticlimactically.
No he said in a long drawl, I cant figure it out. There should be a way, but I cant figure it out.
What?! Why were you in the Company if you cant even work through a little probability problem? pestered Gambor.
Alright, then. said Mathos quietly, You do it.
Gambor stuttered, Wh-what? Im your lieutenant; I havent done this stuff for years!
Dont look at me, joked Boffee, Im just the slinger.
Im just the engineer, murmured Mimbi. Numbers arent my thing, but no one heard him.
Hmph, said Turi finally. Well, I cant figure it out, but I know you all are going about this in completely the wrong way. Everyone turned to stare at him. There are only two hat colors, a binary decision. There ought to be a way to apply the rules of Balinean algebra in our favor. Youre forgetting one crucial piece of information. Each color hat may well be an independent probability, but that doesnt mean our answers have to be!
Dont you mean guesses, scoffed Gambor.
No! replied Turi happily, I mean answers. I dont believe we need to guess at all! I believe there is a way to assure that each and every one of us walks out of here alive! Well perhaps not all of us Mimbi saw Turis eyes meet his own for just a split second before Turi looked away. You see, well know what each dwarf before us answered!
So what? asked Mathos. Well know that they guessed something that has nothing to do with what our own hat colors are.
It will if they base their answer off of the colors of the hats in front of them! squealed Turi in triumph.
Hmm mused Mathos, I see what you might be getting at, but if each answer is based on the hats that can be seen, how is anyone going to be able to answer which hat color theyve so cleverly determined is theirs, from the previous answers? Would only I, as the tallest dwarf here, be free to answer what I know to be my hat color?
Well jumped in Mimbi, now excited by the puzzle, what if each of us answered, from previous answers, what must be our hat color, but we gave the answer in a different way, based on the hats in front of us?
Hmm said Turi.
Gambor asked, What do you mean, a different way? We wont be able to hear each other give the answers we whisper it to the game-master. Is he going to mimic the way we whisper it to him?
Maybe. It could be one of those Balseraph mimics. joked Boffee. Mimbi chortled.
We cant rely on that, said Mathos, not amused.
Ah! cried Gambor, Ive got it! We only need to be able to whisper loud enough for the person in front of us to hear us. If our hat color, as determined from the person behind us, is different from the hat color in front of us, then we whisper our answer to the game-master quietly, so that only he can hear. But if its the same color, we whisper it just loudly enough that Perpentach doesnt realize the person in front of us can hear our answer! Turi, youre a genius!
No. said Mathos sternly. That is unacceptable. The game-master could very easily decide we had whispered too loudly and kill us all. Besides, what if the crowd is noisy? Theres no guarantee any whisper could be overheard. Were not even sure how close together in line well be.
Mimbi suddenly realized there was another problem with Gambors plan, and he had a sneaking feeling that any plan they might come up with for the rest of the night would have a similar drawback.
Aha! No, wait Hmm said Turi, and everyone looked back at him. Everybody, get in single file line, in order from tallest to shortest. Huddle together as close as you can; I need to test the extreme case.
Everyone blinked in shock, until Mathos shouted, You heard the man, get a move on! and they were all lined up in the blink of an eye. Mathos was at the front, with Gambor behind him, with Boffee behind him, and with Turi sandwiched between him and finally Mimbi.
Now, then, wheezed Turi. He really was sandwiched. Can everybody see the heads of all the dwarves in front of him? Everyone answered in the affirmative. Perfect! The solution is similar to a bit of clever work I did on the latest clockwork golem design, to ensure that all the parts were interacting smoothly
Can we please get out of line now?! gasped Boffee.
Hmm? Oh, yes, of course you can. Why not?
All except Turi sat back down, Gambor and Boffee with more than a bit of relief.
Alright, now continued Turi, the trick is to convert the colors into binary numbers. Intuitively, lets take black to be zero and white to be unity.
What are you babbling about? pestered Gambor, You get black by combining all colors it should be unity! Mathos sighed.
Hmm? said Turi. Oh, yes, I suppose you could look at it that way, if you are dealing with reflection off colored surfaces. I was thinking about light directly. But very well, lets take black to be
Black is zero. White is one. Get on with it, interrupted Mathos.
Yes. As I was saying, its quite ingenious. You see, when you want parts of the golem to communicate with one another, you have one part send a set of signals to the other, comprised of magical current. Either a given signal has current (one) or it doesnt (zero). But occasionally any given magical signal can spontaneously switch itself on or off. It doesnt happen often, but it happens often enough that if we dont control for it, bad things can happen. Do you remember the Morrtebio Brewery incident a couple years ago?
What Morrtebio Brewery? joked Boffee. Mimbi chortled. Mathos sighed.
Anyways, heres the ingenious part: you relegate the data you want to transmit to certain elements of your signal set, and you assign the remaining signals either one or zero in a controlled way, so as to ensure that the sum of all the numbers (zero or one) of the signal set is even. We call it a parity check. Each golem part is set to re-request a signal set it receives if its signals add up to an odd number. Actually, Ive been musing on it for the past couple years and I think I can make that more efficient
Mathos frowned, Is any of this actually relevant to our current predicament?
Hmm? said Turi. Oh, no, I was just saying that the solution to our predicament reminded me of parity. Mathos sighed. You see, as I verified with the line, each of us will certainly be able to see all the hats in front of us. Convert their colors to numbers, as we had previously decided, and sum them up. There is either an even number or an odd number. All you need to know is whether there are an even number or odd number in front of the person behind you! If it matches yours, then you yourself must be zero, or black. If its different, then you must be white!
Mathos looked forlorn, And just how will we know the parity the person behind us sees?
Turi looked up, straight at Mimbi, Well that That rests on the shoulders of the dwarf who can see everyones everyone elses hats Everyone turned to look at Mimbi. Im sorry, my lad. I cant think of any way for you to know your own hat color. So your answer must tell the rest of us how many white hats you see. Answer white if the number is even. Answer black if the number is odd. Can you remember that? The rest of us just have to keep track of what colors have been said and subtract them off.
Everyone was silent for some time.
Boffee finally broke the silence, Hey, think of it this way: you might just say your hat color, and then every one of us will get to walk out of here!
Can you do it, lad? Mathos asked, Our lives depend on you.
Mimbi swallowed. White if its even. Black if its odd. I can do it. And then he smiled a little. Being short had its disadvantages. But at least, if he died, hed die saving everyone elses life. He just kept reminding himself of that as he tried to fall asleep. He tried not to think about it being his last night alive.
* * *
* * *
Mimbi had a deep, restful sleep, and Gambor had to shake him awake when their prison guard opened the door. Mimbi opened his eyes and saw Gambors filled with tears.
You can do it, lad, he whispered.
Mimbi slowly remembered what the discussion had been about last night. A ripple of fear seized his heart for a moment, but he reminded himself that he would be remembered as a hero. White if its even. Black if its odd. He nodded, and sat up.
Everyone else was already standing. They knew what was happening, but Boffee had the sense to continue the charade for a little while longer. Wheres Aila? Wheres our food? he asked the guard.
The guard grinned. Youre not eating this morning. The Momus has summoned you to his arena for a little game. And youll be going blindfolded. At this, he pulled out five black strips of cloth from his pocket and wrapped one around each dwarfs head in turn. Then he had each dwarf put a hand on the shoulder of another. By the height difference, Mimbi knew that he was behind Turi, and he suspected they were already in the same order they had been last night. So far, so good. White if its even. Black if its odd.
They were marched single-file among passageway after passageway filled with the moans of prisoners and growls of captured animals, all steadily uphill. Finally, they entered a hallway which had sunlight at the far end. Mimbi couldnt see a thing, but he felt it on his face, and never had it felt so good. The Luchiurp people werent like their cousins the Khazad. They needed to be outdoors, at least occasionally. It was like the touch of Nantosuelta, a caress of hope. But the ripple of fear came back for just a moment. He fought it back, telling himself the hope was not for him, but for his four companions. Black if its even. White if its odd. No, that wasnt right. Black if its odd. White if its even.
Mimbi could tell they had just left the passageway, and that they really were outside now. Presumably this was Perpentachs arena. The place of his death, for he had consigned himself to that. Whether he might live or not was of no concern. All he had to do was answer black if its odd, and white if its even.
If whats odd? If whats even? he thought in a momentary panic. He had to rack his brains through the conversation last night to remember. The number of white hats. If he counted an odd number of white hats, he said black. If he counted an even number of white hats, he said white. Or was it black hats he was supposed to count? What if I count the wrong color! But then he thought it through, and realized it wouldnt matter. He would be looking at four hats: if there were an even number of white hats, thered be an even number of black hats, and vice versa. That calmed him down. Black if its odd. White if its even.
They were marched to the middle of the arena, and could hear a huge crowd gathered in the stands around them. There was all the hustle-bustle one might expect of an audience, before the show started, but then Perpentach himself, from his regal balcony, began to speak:
Come one, come all
To festivitys hall
And ere you go home
Watch each little gnome
Fight for its name
In our very own Circus Game!
The midgets are blindfolded
But soon they will see
The gnome just in front
Is just taller than he
If any gnome speaks
Or moves just an inch
The game is thus ended
All die in a pinch
My harlequin Fred
Will set on each head
A hat of a color
Not known by he under
One by one all, from smallest to tall
Will whisper to Fred the shade on his head
The audience will hear what passed through Freds ear
But if he is wrong, the gnome wont last long
His head will be freed
From his torso and knees
Yet if he is right
He may go as he please
Mimbi felt the tight cap being squeezed onto his head, and then the blindfold being taken off. He could easily see the hats of each dwarf in front of him. Mathos, all the way in front, had a white hat. Gambor, just behind, had a black hat. Boffee, too had a black hat. And Turi had a white hat. Two white hats and two black hats. White if its even. Simple.
But that wasnt all Mimbi could see. He also saw Fred the harlequin, with an unbelievably huge grin on his face. And Mimbi knew why. Because pinned to Freds livery was a silver medallion, which Fred was holding at a very specific angle. Mimbi could see himself reflected perfectly in the metal, and he was wearing a black hat.
* * *
* * *
All the blood drained from Mimbis face. He felt his limbs about to buckle. White if its even. White if its even! he tried to tell himself. When Fred bent his ear to Mimbis mouth, Mimbi couldnt help but squeak out: Black.
A burly pair of arms snatched Mimbi from behind and covered his mouth so that he couldnt make any more sound if he wanted to. He watched helplessly as the Game spiraled beyond his control.
* * *
BLACK! Turi heard behind him, and he thought quickly: I see an odd number of white hats. So did Mimbi. Therefore I am wearing a black hat. Trivial! And then, The crowd doesnt seem to have made much noise. I do believe young Mimbi was lucky enough to have also been wearing a black hat!
Fred the Harlequin folded himself at the waist, as if in a regal bow. Turi wondered why he would waste so much time with pleasantries, and when the harlequins ear was at Turis mouth, Turi confidently whispered Black, with a smile.
Turi gasped in shock when he felt the cool metal blade touch the nape of his neck. He made no more sound when it went through his windpipe.
* * *
BLACK! Boffee heard behind him. But Boffee had also heard what distinctly sounded like Turi gasping. Boffee had to resist every impulse in his body as he nearly turned around. It sounded like Turi had been killed. But then, that would mean his hat had actually been white. Unless Aila had lied to them? Impossible! She would never do such a thing. But, if Boffees hat had been white, and Mimbi had seen an an odd number, then Ah! Boffee couldnt keep track of it all. But here was Fred the Harlequins ear already at Boffees mouth.
Wait, Boffee hissed. He never felt the thin-wire garrote slice off his head.
* * *
WHITE! Gambor heard behind him. He could hardly believe it. It was working. There were an odd number of white hats, and only one of them had so far had a white hat other than perhaps the poor lad Mimbi; he would be remembered. Gambor saw a white hat in front of him, so there had to have been three white hats at the outset (not counting poor Mimbi, if his happened to be white too).
Fred the Harlequin leaned down a little, and Gambor triumphantly whispered into the fools ear, White. Then Gambor screamed, when Fred disemboweled him with a knife.
* * *
WHITE! Mathos heard behind him, and then Gambor screamed and the crowd roared with laughter. Mathos understood what had happened. Of course the Clown King would lay this trap. Everything was a lie. Or perhaps nothing was, and they had been done in by their own overconfidence. He suspected that not one dwarf behind him had survived, despite the relative peace of the crowd til now. They were all at the Clown Kings bidding.
Fred the Harlequin circled around to face Mathos. Give your answer to the crowds yourself, good Captain! sang the jester. His hand still held the knife, dripping in Gambors blood. Mathos felt his rage boiling, and then he unleashed it.
RED! he roared, and jumped at Freds throat. Fred stabbed him in the belly once, twice, but nothing could keep Captain Mathoss hands from cracking poor Freds windpipe. The crowd went wild with glee, and then cheered even louder as three spearmen rushed in and impaled Mathos, over and over. But Mathos ignored them, as all his blood leaked from his body. He focused on Freds pale painted face, and told himself over and over that it was Perpentach himself, until he let himself die.
* * *
* * *
The hands finally let loose their grip on Mimbis head, but they wouldnt let him go. Mimbi howled in pain. I killed them all. I killed them all. No other thought would fit through his head. I killed them all.
After some minutes, the crowds antics died down and Perpentach addressed Mimbi with his accursed rhymes.
Your captain broke the rules
When he jumped on poor Fred
By my stated guidelines
You should be dead
But a king needs know mercy
And I know it well
So be free, little dwarf
And do this tale tell
Mimbi was finally released, and in a daze started wandering towards the large entrance to the arena. I killed them all. I killed them all. When he reached the gate, he looked behind him, to Perpentachs balcony. Just next to the Clown King, he could see Aila, dressed in a courtesans finery. She noticed him, smiled sweetly, and waved farewell.
Khazad dwarves are perfectly comfortable deep inside the rock, but their distant Luchiurp brethren prefer their above-ground workshops - especially when they're stuck in a wet, mouldy, bug-infested, Balseraph dungeon! Prisoners of a war they never asked for, Captain Mathos and company are stuck waiting for something - anything - to happen...
-----------------------
Notes
Hullo. This story doesn't track any particular game; it just takes place in Kael's Fall from Heaven universe. I'm assuming this is the right place to put it..?
If you are familiar with the mod, many characters and events will be familiar to you, but I *think* there's enough context in the story for it to stand on its own. If anyone gets confused on any given point and thinks the story should flesh out any particular detail, let me know and I'll see if I can tweak things.
Also, if you are more familiar with the universe than I am, let me know if any details just aren't right - I do want this story to be perfectly consistent with Kael's.
And of course alert me to any typos. I speak English well, but I don't type so good.
I've split it up into multiple spoiler tags just to improve readability a little bit. Let me know if you like it!
-----------------------
The Circus Game
Spoiler :
He heard it again. The far-off stamping of hooves. His own raggedy breath as he struggled to keep up with Captain Mathos and the Company. Then the shouts of the savage horsemen as another contingent no one had seen coming suddenly appeared at the Companys flank. They corralled the dwarves, who now routed, running every which way to avoid the huge nets the Hippus mercenaries were throwing on them.
But Mimbi himself was small, even for a dwarf. They came at him with little butterfly nets. Each horseman took turns swinging his net, until finally one caught the tip of Mimbis beard. Oh, how it scratched the beard!
Not the beard! Not the beard! Mimbi awoke with a start.
Gambor looked up, annoyed, and squinted in the dim light. Its just a spider, he said.
Mimbi threw himself up and raked vigorously at his beard with his fingers. A little spider fell out. On its back was a yellow and red pattern that looked for all the world like a grinning mouth, teeth and all. Mimbi tried stamping on it, but it scampered away into a little hole in the wall.
Dreaming again? Mathos said dryly.
Mimbi plopped back down, embarrassed, onto the cold damp stone.
Its alright. Ailall be here with dinner soon, said Boffee.
Probably mouldy bread and cold soup again, grumbled Mimbi.
Better than nothing, replied Mathos. Mimbi decided to stay quiet. It always seemed like the others were short with him, but what else would you expect after two weeks in a cramped, smelly, wet dungeon? Of course, he always seemed short to them.
Oh, great. Their hosts silly sense of humor was rubbing itself off on him through the six-foot thick stone walls. That was Mimbis job. Masonry. He could tell how thick or hollow a wall was just by rapping on it with a knuckle. Mathos had been glad to have him, back when the Companys job was just fortifying a little suburb of Ithralia. Then the mercenaries started raiding, because some crazy Illian thought his soul was trapped in Barnaxus. When the old golem had heard this, he marched onto a hill three miles from Ithralia and gave himself up to the Hippus raiders. But the savages thought they could sell of living Luchiurp to an even crazier customer, so they kept raiding. And now here was all that was left of Mathoss Company: five dwarves in the dungeon of Perpentach, the Momus, the Clown King.
Mimbi felt the stone shudder ever so slightly. Turi, wake up! Dinners on its way. The old dwarf just rolled over, but Gambor shook him until he snorted and yawned.
Hmm? Turi said.
But by then the prison guard was perfectly audible on the other side of the iron door, keys jangling. The door opened with a screech for just a moment, and in stumbled a scrawny human girl dressed in dirty rags and carrying a tray of bread just bread.
Aila, you look well, said Boffee, ever the charmer. He probably hadnt even noticed yet how meager their rations were today. Mimbis stomach growled just looking at it.
Mathos noticed. Even when he had lost everything, he still looked out for his own. Is this all you have for us today, Aila? he asked sadly.
Aila knelt down to the floor so that she was eye-level with Mathos, who had stood up when she entered. She set the tray down and handed out the mouldy (of course ) loaves to each dwarf.
No, she whispered hurriedly, I bring news as well. She glanced warily behind her, to make sure the door was shut and the guard wouldnt overhear. Perpentach has taken Beeri Bawl captive.
Every dwarf in the chamber moaned. The Luchiurp king had been captured. All hope was lost.
Quiet! Aila screamed in a hoarse whisper. Theyll She glanced back at the door, and then continued, even quieter, Theyll hurt me if they find out I told you.
Mathos nodded sagely. We appreciate the risk youve taken. You have our gratitude, he said as quietly as his deep booming voice could be made to.
But Aila shook her head. Thats not all. Perpentach is throwing a festival in celebration tomorrow
Isnt he always? joked Boffee, and Mimbi chortled a little.
Aila continued, where the main event is a game, and you five are the contestants.
The dwarves were all quiet now. They listened intently as Aila explained, Youll be blindfolded, marched into the arena, and lined up, from tallest to shortest. Each of you will get a hat, and then youll be unblindfolded. Perpentach will explain the rules: if any of you try to turn or speak or do anything youre not supposed to, youll all be beheaded on the spot. Youll only be able to see the back of the dwarf in front of you and I guess maybe the heads of the dwarves in front of him. The game-master starts in the back of the line, with the shortest dwarf, and goes on one by one to the tallest. The contest is to guess what color hat youre wearing, and whisper it to the game-master, who announces your answer to the crowd. If youre wrong, hell kill you. But in the end, Perpentach will release any of you who guessed right!
Aila glanced behind her yet again, and lowered her voice even more, But heres the part Perpentach wont tell you tomorrow: all the hats will either be black or white!
Hmm said Turi.
I dont underst Gambor started, but he hushed when the door suddenly screamed open.
Why havent you knocked yet?! yelled the guard to Aila.
Aila scrambled up with the tray, keeping her head down. Im sorry, sir. I was just seizing a bit of rest. I know youll work me so oh! as the prisoner spanked her bottom. They both disappeared behind the door as it shut with a resounding boom.
After a moment, Boffee lamented, Oh, that we could all escape! Alas, it is not to be.
Mimbi added hopefully, Well, thanks to Aila, I guess we each have a fifty-fifty shot.
Hmm said Turi.
I dont understand, said Gambor, Why would Perpentach agree to release any of us? This must be some sort of trick.
Mathos shook his head. The Clown King is mad. He has been known to do such things in the past. And strange as it may seem, hes never been one to break his word. Thats not his sort of trickery. No his trickery is this illusory hope, that each of us might just escape. Itll make our deaths so much sweeter to his audience.
But why would he only put black or white hats on us? Mimbi asked.
Mathos snorted, To make it all the more amusing to the audience when we try to guess ridiculous colors like lavender, or chartreuse.
Boffee frowned. But now we just guess black or white? Perpentach will know someone told us Aila will be punished severely.
Mathos nodded sadly, I imagine you are right. But we cannot waste her gift to us by consigning ourselves to certain death. Boffee nodded.
But Mimbi himself was small, even for a dwarf. They came at him with little butterfly nets. Each horseman took turns swinging his net, until finally one caught the tip of Mimbis beard. Oh, how it scratched the beard!
Not the beard! Not the beard! Mimbi awoke with a start.
Gambor looked up, annoyed, and squinted in the dim light. Its just a spider, he said.
Mimbi threw himself up and raked vigorously at his beard with his fingers. A little spider fell out. On its back was a yellow and red pattern that looked for all the world like a grinning mouth, teeth and all. Mimbi tried stamping on it, but it scampered away into a little hole in the wall.
Dreaming again? Mathos said dryly.
Mimbi plopped back down, embarrassed, onto the cold damp stone.
Its alright. Ailall be here with dinner soon, said Boffee.
Probably mouldy bread and cold soup again, grumbled Mimbi.
Better than nothing, replied Mathos. Mimbi decided to stay quiet. It always seemed like the others were short with him, but what else would you expect after two weeks in a cramped, smelly, wet dungeon? Of course, he always seemed short to them.
Oh, great. Their hosts silly sense of humor was rubbing itself off on him through the six-foot thick stone walls. That was Mimbis job. Masonry. He could tell how thick or hollow a wall was just by rapping on it with a knuckle. Mathos had been glad to have him, back when the Companys job was just fortifying a little suburb of Ithralia. Then the mercenaries started raiding, because some crazy Illian thought his soul was trapped in Barnaxus. When the old golem had heard this, he marched onto a hill three miles from Ithralia and gave himself up to the Hippus raiders. But the savages thought they could sell of living Luchiurp to an even crazier customer, so they kept raiding. And now here was all that was left of Mathoss Company: five dwarves in the dungeon of Perpentach, the Momus, the Clown King.
Mimbi felt the stone shudder ever so slightly. Turi, wake up! Dinners on its way. The old dwarf just rolled over, but Gambor shook him until he snorted and yawned.
Hmm? Turi said.
But by then the prison guard was perfectly audible on the other side of the iron door, keys jangling. The door opened with a screech for just a moment, and in stumbled a scrawny human girl dressed in dirty rags and carrying a tray of bread just bread.
Aila, you look well, said Boffee, ever the charmer. He probably hadnt even noticed yet how meager their rations were today. Mimbis stomach growled just looking at it.
Mathos noticed. Even when he had lost everything, he still looked out for his own. Is this all you have for us today, Aila? he asked sadly.
Aila knelt down to the floor so that she was eye-level with Mathos, who had stood up when she entered. She set the tray down and handed out the mouldy (of course ) loaves to each dwarf.
No, she whispered hurriedly, I bring news as well. She glanced warily behind her, to make sure the door was shut and the guard wouldnt overhear. Perpentach has taken Beeri Bawl captive.
Every dwarf in the chamber moaned. The Luchiurp king had been captured. All hope was lost.
Quiet! Aila screamed in a hoarse whisper. Theyll She glanced back at the door, and then continued, even quieter, Theyll hurt me if they find out I told you.
Mathos nodded sagely. We appreciate the risk youve taken. You have our gratitude, he said as quietly as his deep booming voice could be made to.
But Aila shook her head. Thats not all. Perpentach is throwing a festival in celebration tomorrow
Isnt he always? joked Boffee, and Mimbi chortled a little.
Aila continued, where the main event is a game, and you five are the contestants.
The dwarves were all quiet now. They listened intently as Aila explained, Youll be blindfolded, marched into the arena, and lined up, from tallest to shortest. Each of you will get a hat, and then youll be unblindfolded. Perpentach will explain the rules: if any of you try to turn or speak or do anything youre not supposed to, youll all be beheaded on the spot. Youll only be able to see the back of the dwarf in front of you and I guess maybe the heads of the dwarves in front of him. The game-master starts in the back of the line, with the shortest dwarf, and goes on one by one to the tallest. The contest is to guess what color hat youre wearing, and whisper it to the game-master, who announces your answer to the crowd. If youre wrong, hell kill you. But in the end, Perpentach will release any of you who guessed right!
Aila glanced behind her yet again, and lowered her voice even more, But heres the part Perpentach wont tell you tomorrow: all the hats will either be black or white!
Hmm said Turi.
I dont underst Gambor started, but he hushed when the door suddenly screamed open.
Why havent you knocked yet?! yelled the guard to Aila.
Aila scrambled up with the tray, keeping her head down. Im sorry, sir. I was just seizing a bit of rest. I know youll work me so oh! as the prisoner spanked her bottom. They both disappeared behind the door as it shut with a resounding boom.
After a moment, Boffee lamented, Oh, that we could all escape! Alas, it is not to be.
Mimbi added hopefully, Well, thanks to Aila, I guess we each have a fifty-fifty shot.
Hmm said Turi.
I dont understand, said Gambor, Why would Perpentach agree to release any of us? This must be some sort of trick.
Mathos shook his head. The Clown King is mad. He has been known to do such things in the past. And strange as it may seem, hes never been one to break his word. Thats not his sort of trickery. No his trickery is this illusory hope, that each of us might just escape. Itll make our deaths so much sweeter to his audience.
But why would he only put black or white hats on us? Mimbi asked.
Mathos snorted, To make it all the more amusing to the audience when we try to guess ridiculous colors like lavender, or chartreuse.
Boffee frowned. But now we just guess black or white? Perpentach will know someone told us Aila will be punished severely.
Mathos nodded sadly, I imagine you are right. But we cannot waste her gift to us by consigning ourselves to certain death. Boffee nodded.
Spoiler :
So what do we do? asked Mimbi, Do we all just say black and guarantee that half of us get out?
Guarantee? snorted Gambor derisively. I suppose youre sure the Clown King will put three black hats and two white hats down? The man is mad! We cant be sure of anything!
Hmm said Turi.
Oh thought Mimbi. I guess I just assumed theyd be randomly distributed. You know, on average, half would be black and half would be white.
Boffee pointed out, With an even distribution. And even if that is the case, there arent enough of us to rely on that. It isnt beyond the realm of possibility for all five hats to be randomly assigned the same color. Then he frowned again. Blimey, I cant remember my Posi Distributions. Being in the Companys rotted my brains.
Mathos scowled at him, Thats not even a Posi Distribution problem. Theres a one in thirty-two chance that all the hats would be black. Theres a lets see ten in thirty-two chance that three of the hats are black.
Boffee grinned, chagrined, but Gambor picked up the conversation, And a ten in thirty-two chance that only two are black. But none of that matters, because we dont have any reason to think therell be an even, random distribution!
Mimbi countered, But we dont have the slightest idea what else it could be, either, so we might as well just assume it is.
So we all just say black, then? Boffee asked. Would it be better for each of us to just randomly decide on our own whether to say black or white?
Hmm said Turi.
Mimbi frowned. That didnt seem right. Well we know that (assuming an even random distribution) the highest probability is that there are two or three black hats. So wed definitely want to make sure at least two or three of us say black. And since we dont know which of us might be black, wouldnt it be best for all of us to say it? Itd be a little embarrassing if everyone with a black hat accidentally said white, and everyone with a white hat accidentally said black, dont ya think?
Mathos sighed, That sounds like it could use a mathematical proof to back it up. Turi, youre the expert here. I dont suppose you could tell us which strategy gives the best chance of getting the most of us out of here?
Turi breathed in audibly, then let it out anticlimactically.
No he said in a long drawl, I cant figure it out. There should be a way, but I cant figure it out.
What?! Why were you in the Company if you cant even work through a little probability problem? pestered Gambor.
Alright, then. said Mathos quietly, You do it.
Gambor stuttered, Wh-what? Im your lieutenant; I havent done this stuff for years!
Dont look at me, joked Boffee, Im just the slinger.
Im just the engineer, murmured Mimbi. Numbers arent my thing, but no one heard him.
Hmph, said Turi finally. Well, I cant figure it out, but I know you all are going about this in completely the wrong way. Everyone turned to stare at him. There are only two hat colors, a binary decision. There ought to be a way to apply the rules of Balinean algebra in our favor. Youre forgetting one crucial piece of information. Each color hat may well be an independent probability, but that doesnt mean our answers have to be!
Dont you mean guesses, scoffed Gambor.
No! replied Turi happily, I mean answers. I dont believe we need to guess at all! I believe there is a way to assure that each and every one of us walks out of here alive! Well perhaps not all of us Mimbi saw Turis eyes meet his own for just a split second before Turi looked away. You see, well know what each dwarf before us answered!
So what? asked Mathos. Well know that they guessed something that has nothing to do with what our own hat colors are.
It will if they base their answer off of the colors of the hats in front of them! squealed Turi in triumph.
Hmm mused Mathos, I see what you might be getting at, but if each answer is based on the hats that can be seen, how is anyone going to be able to answer which hat color theyve so cleverly determined is theirs, from the previous answers? Would only I, as the tallest dwarf here, be free to answer what I know to be my hat color?
Well jumped in Mimbi, now excited by the puzzle, what if each of us answered, from previous answers, what must be our hat color, but we gave the answer in a different way, based on the hats in front of us?
Hmm said Turi.
Gambor asked, What do you mean, a different way? We wont be able to hear each other give the answers we whisper it to the game-master. Is he going to mimic the way we whisper it to him?
Maybe. It could be one of those Balseraph mimics. joked Boffee. Mimbi chortled.
We cant rely on that, said Mathos, not amused.
Ah! cried Gambor, Ive got it! We only need to be able to whisper loud enough for the person in front of us to hear us. If our hat color, as determined from the person behind us, is different from the hat color in front of us, then we whisper our answer to the game-master quietly, so that only he can hear. But if its the same color, we whisper it just loudly enough that Perpentach doesnt realize the person in front of us can hear our answer! Turi, youre a genius!
Spoiler :
No. said Mathos sternly. That is unacceptable. The game-master could very easily decide we had whispered too loudly and kill us all. Besides, what if the crowd is noisy? Theres no guarantee any whisper could be overheard. Were not even sure how close together in line well be.
Mimbi suddenly realized there was another problem with Gambors plan, and he had a sneaking feeling that any plan they might come up with for the rest of the night would have a similar drawback.
Aha! No, wait Hmm said Turi, and everyone looked back at him. Everybody, get in single file line, in order from tallest to shortest. Huddle together as close as you can; I need to test the extreme case.
Everyone blinked in shock, until Mathos shouted, You heard the man, get a move on! and they were all lined up in the blink of an eye. Mathos was at the front, with Gambor behind him, with Boffee behind him, and with Turi sandwiched between him and finally Mimbi.
Now, then, wheezed Turi. He really was sandwiched. Can everybody see the heads of all the dwarves in front of him? Everyone answered in the affirmative. Perfect! The solution is similar to a bit of clever work I did on the latest clockwork golem design, to ensure that all the parts were interacting smoothly
Can we please get out of line now?! gasped Boffee.
Hmm? Oh, yes, of course you can. Why not?
All except Turi sat back down, Gambor and Boffee with more than a bit of relief.
Alright, now continued Turi, the trick is to convert the colors into binary numbers. Intuitively, lets take black to be zero and white to be unity.
What are you babbling about? pestered Gambor, You get black by combining all colors it should be unity! Mathos sighed.
Hmm? said Turi. Oh, yes, I suppose you could look at it that way, if you are dealing with reflection off colored surfaces. I was thinking about light directly. But very well, lets take black to be
Black is zero. White is one. Get on with it, interrupted Mathos.
Yes. As I was saying, its quite ingenious. You see, when you want parts of the golem to communicate with one another, you have one part send a set of signals to the other, comprised of magical current. Either a given signal has current (one) or it doesnt (zero). But occasionally any given magical signal can spontaneously switch itself on or off. It doesnt happen often, but it happens often enough that if we dont control for it, bad things can happen. Do you remember the Morrtebio Brewery incident a couple years ago?
What Morrtebio Brewery? joked Boffee. Mimbi chortled. Mathos sighed.
Anyways, heres the ingenious part: you relegate the data you want to transmit to certain elements of your signal set, and you assign the remaining signals either one or zero in a controlled way, so as to ensure that the sum of all the numbers (zero or one) of the signal set is even. We call it a parity check. Each golem part is set to re-request a signal set it receives if its signals add up to an odd number. Actually, Ive been musing on it for the past couple years and I think I can make that more efficient
Mathos frowned, Is any of this actually relevant to our current predicament?
Hmm? said Turi. Oh, no, I was just saying that the solution to our predicament reminded me of parity. Mathos sighed. You see, as I verified with the line, each of us will certainly be able to see all the hats in front of us. Convert their colors to numbers, as we had previously decided, and sum them up. There is either an even number or an odd number. All you need to know is whether there are an even number or odd number in front of the person behind you! If it matches yours, then you yourself must be zero, or black. If its different, then you must be white!
Mathos looked forlorn, And just how will we know the parity the person behind us sees?
Turi looked up, straight at Mimbi, Well that That rests on the shoulders of the dwarf who can see everyones everyone elses hats Everyone turned to look at Mimbi. Im sorry, my lad. I cant think of any way for you to know your own hat color. So your answer must tell the rest of us how many white hats you see. Answer white if the number is even. Answer black if the number is odd. Can you remember that? The rest of us just have to keep track of what colors have been said and subtract them off.
Everyone was silent for some time.
Boffee finally broke the silence, Hey, think of it this way: you might just say your hat color, and then every one of us will get to walk out of here!
Can you do it, lad? Mathos asked, Our lives depend on you.
Mimbi swallowed. White if its even. Black if its odd. I can do it. And then he smiled a little. Being short had its disadvantages. But at least, if he died, hed die saving everyone elses life. He just kept reminding himself of that as he tried to fall asleep. He tried not to think about it being his last night alive.
* * *
Spoiler :
* * *
Mimbi had a deep, restful sleep, and Gambor had to shake him awake when their prison guard opened the door. Mimbi opened his eyes and saw Gambors filled with tears.
You can do it, lad, he whispered.
Mimbi slowly remembered what the discussion had been about last night. A ripple of fear seized his heart for a moment, but he reminded himself that he would be remembered as a hero. White if its even. Black if its odd. He nodded, and sat up.
Everyone else was already standing. They knew what was happening, but Boffee had the sense to continue the charade for a little while longer. Wheres Aila? Wheres our food? he asked the guard.
The guard grinned. Youre not eating this morning. The Momus has summoned you to his arena for a little game. And youll be going blindfolded. At this, he pulled out five black strips of cloth from his pocket and wrapped one around each dwarfs head in turn. Then he had each dwarf put a hand on the shoulder of another. By the height difference, Mimbi knew that he was behind Turi, and he suspected they were already in the same order they had been last night. So far, so good. White if its even. Black if its odd.
They were marched single-file among passageway after passageway filled with the moans of prisoners and growls of captured animals, all steadily uphill. Finally, they entered a hallway which had sunlight at the far end. Mimbi couldnt see a thing, but he felt it on his face, and never had it felt so good. The Luchiurp people werent like their cousins the Khazad. They needed to be outdoors, at least occasionally. It was like the touch of Nantosuelta, a caress of hope. But the ripple of fear came back for just a moment. He fought it back, telling himself the hope was not for him, but for his four companions. Black if its even. White if its odd. No, that wasnt right. Black if its odd. White if its even.
Mimbi could tell they had just left the passageway, and that they really were outside now. Presumably this was Perpentachs arena. The place of his death, for he had consigned himself to that. Whether he might live or not was of no concern. All he had to do was answer black if its odd, and white if its even.
If whats odd? If whats even? he thought in a momentary panic. He had to rack his brains through the conversation last night to remember. The number of white hats. If he counted an odd number of white hats, he said black. If he counted an even number of white hats, he said white. Or was it black hats he was supposed to count? What if I count the wrong color! But then he thought it through, and realized it wouldnt matter. He would be looking at four hats: if there were an even number of white hats, thered be an even number of black hats, and vice versa. That calmed him down. Black if its odd. White if its even.
They were marched to the middle of the arena, and could hear a huge crowd gathered in the stands around them. There was all the hustle-bustle one might expect of an audience, before the show started, but then Perpentach himself, from his regal balcony, began to speak:
Come one, come all
To festivitys hall
And ere you go home
Watch each little gnome
Fight for its name
In our very own Circus Game!
The midgets are blindfolded
But soon they will see
The gnome just in front
Is just taller than he
If any gnome speaks
Or moves just an inch
The game is thus ended
All die in a pinch
My harlequin Fred
Will set on each head
A hat of a color
Not known by he under
One by one all, from smallest to tall
Will whisper to Fred the shade on his head
The audience will hear what passed through Freds ear
But if he is wrong, the gnome wont last long
His head will be freed
From his torso and knees
Yet if he is right
He may go as he please
Mimbi felt the tight cap being squeezed onto his head, and then the blindfold being taken off. He could easily see the hats of each dwarf in front of him. Mathos, all the way in front, had a white hat. Gambor, just behind, had a black hat. Boffee, too had a black hat. And Turi had a white hat. Two white hats and two black hats. White if its even. Simple.
But that wasnt all Mimbi could see. He also saw Fred the harlequin, with an unbelievably huge grin on his face. And Mimbi knew why. Because pinned to Freds livery was a silver medallion, which Fred was holding at a very specific angle. Mimbi could see himself reflected perfectly in the metal, and he was wearing a black hat.
* * *
Spoiler :
* * *
All the blood drained from Mimbis face. He felt his limbs about to buckle. White if its even. White if its even! he tried to tell himself. When Fred bent his ear to Mimbis mouth, Mimbi couldnt help but squeak out: Black.
A burly pair of arms snatched Mimbi from behind and covered his mouth so that he couldnt make any more sound if he wanted to. He watched helplessly as the Game spiraled beyond his control.
* * *
BLACK! Turi heard behind him, and he thought quickly: I see an odd number of white hats. So did Mimbi. Therefore I am wearing a black hat. Trivial! And then, The crowd doesnt seem to have made much noise. I do believe young Mimbi was lucky enough to have also been wearing a black hat!
Fred the Harlequin folded himself at the waist, as if in a regal bow. Turi wondered why he would waste so much time with pleasantries, and when the harlequins ear was at Turis mouth, Turi confidently whispered Black, with a smile.
Turi gasped in shock when he felt the cool metal blade touch the nape of his neck. He made no more sound when it went through his windpipe.
* * *
BLACK! Boffee heard behind him. But Boffee had also heard what distinctly sounded like Turi gasping. Boffee had to resist every impulse in his body as he nearly turned around. It sounded like Turi had been killed. But then, that would mean his hat had actually been white. Unless Aila had lied to them? Impossible! She would never do such a thing. But, if Boffees hat had been white, and Mimbi had seen an an odd number, then Ah! Boffee couldnt keep track of it all. But here was Fred the Harlequins ear already at Boffees mouth.
Wait, Boffee hissed. He never felt the thin-wire garrote slice off his head.
* * *
WHITE! Gambor heard behind him. He could hardly believe it. It was working. There were an odd number of white hats, and only one of them had so far had a white hat other than perhaps the poor lad Mimbi; he would be remembered. Gambor saw a white hat in front of him, so there had to have been three white hats at the outset (not counting poor Mimbi, if his happened to be white too).
Fred the Harlequin leaned down a little, and Gambor triumphantly whispered into the fools ear, White. Then Gambor screamed, when Fred disemboweled him with a knife.
* * *
WHITE! Mathos heard behind him, and then Gambor screamed and the crowd roared with laughter. Mathos understood what had happened. Of course the Clown King would lay this trap. Everything was a lie. Or perhaps nothing was, and they had been done in by their own overconfidence. He suspected that not one dwarf behind him had survived, despite the relative peace of the crowd til now. They were all at the Clown Kings bidding.
Fred the Harlequin circled around to face Mathos. Give your answer to the crowds yourself, good Captain! sang the jester. His hand still held the knife, dripping in Gambors blood. Mathos felt his rage boiling, and then he unleashed it.
RED! he roared, and jumped at Freds throat. Fred stabbed him in the belly once, twice, but nothing could keep Captain Mathoss hands from cracking poor Freds windpipe. The crowd went wild with glee, and then cheered even louder as three spearmen rushed in and impaled Mathos, over and over. But Mathos ignored them, as all his blood leaked from his body. He focused on Freds pale painted face, and told himself over and over that it was Perpentach himself, until he let himself die.
* * *
Spoiler :
* * *
The hands finally let loose their grip on Mimbis head, but they wouldnt let him go. Mimbi howled in pain. I killed them all. I killed them all. No other thought would fit through his head. I killed them all.
After some minutes, the crowds antics died down and Perpentach addressed Mimbi with his accursed rhymes.
Your captain broke the rules
When he jumped on poor Fred
By my stated guidelines
You should be dead
But a king needs know mercy
And I know it well
So be free, little dwarf
And do this tale tell
Mimbi was finally released, and in a daze started wandering towards the large entrance to the arena. I killed them all. I killed them all. When he reached the gate, he looked behind him, to Perpentachs balcony. Just next to the Clown King, he could see Aila, dressed in a courtesans finery. She noticed him, smiled sweetly, and waved farewell.
