"I'm set to jump," replied Aggie from the top of the pyramid. "Just one question Mark. I take it you're not too surprised that I haven't seen anybody?"
"No, not too surprised."
"I was. Why do you think the people aren't here?"
"Oh, I think you've found them Aggie. And if we were doing this back in December, I bet you would already be talking to them. But we've been on this planet for almost fourteen weeks. The people from your Party probably have been in the pyramid for almost as long. Fourteen weeks Aggie! I wouldn't be surprised if they gave up watching the boring doors a long time ago."
"Uh… Yeah, good point. I didn't think of that. So what's my mission? If I'm testing for broken rings so efficiently, I'll be out of here before I find them."
"Aggie, that's exactly what I'm hoping for," answered Mark.
"Huh?!"
"Think it through Aggie. Let's figure out how to rescue these folks before we try to do it. For all we know, the broken ring might lead to a spot where…" Mark suddenly felt very uncomfortable. "Well, to where it's not a good spot to be. I know I'm being very blunt, but I think you need to be a pathfinder Aggie. Concentrate on finding your way back home. Then we can return to the pyramid in confidence and in group force and offer Party #4 a clear solution to their imprisonment."
There was a short pause. "I agree entirely," Aggie finally replied. "Setting tumbler code to A-4 and jumping from the physical carriage 1, base ring position zero… I'm at carriage 300 now. The floor is 28, but you already know that. I won't report the floor numbers anymore."
"I have the data too," said Emily, "and Mark sent me his reverse lookup code. Aggie, your position on the base ring is 387. I'm logging A-4 jump size as 387."
"Clicking to A-5 and jumping…" said Aggie. "Carriage 312."
Akiko and Alison linked for a moment, and then Akiko said, "Emily, Mark, why don't you guys take a break? You deserve it. Alison and I see how to run the lookup code. We understand this very well. We will log the jump sizes and check each other's work."
Alison spoke up. "Base ring position is now 348. Logging A-5 jump size as 634."
"Clicking to A-6," replied Aggie. "Carriage 641."
"Ring position is 43," replied Akiko. "A-6 jump size is 368."
"Clicking to A-7," continued Aggie. She settled into working with Akiko and Alison and soon the three women became a very competent and diligent team. Aggie was on her way home.
A few minutes later Mark and Emily shared an affectionate hug at the Hilton, and then they got on another line and started chatting with Fatima back at the Bee Park.
Fatima spoke up. "Mark, mind if I pick your brain for a while?"
Emily added. "Yeah, me too Mark." She leaned over and kissed his cheek. "I want to know how your mind works too."
"Go ahead Fatima," Mark said as he squeezed Emily's waist.
"The original problem, 672 by 673 possibilities, it's been reduced to an almost trivial search. How did you see how to do this?"
"Well, Emily's example of seven elevators was the key. Remember she had a base ring of 1-2-3-4-5-6-7-1? That was her code 1."
"Yes, I remember."
"And code 3 was 1-4-7-3-6-2-5-1. I got to thinking, what if this series was the base ring, the true code 1. Then code 2, making two unit jumps on this base ring would yield 1-7-6-5-4-3-2-1. But that code was the old code 6. Do you see what's going on? Knowing the order of any complete jump code gives you a tremendous amount of information."
He returned another kiss from Emily and then continued. "How many ways can you order seven elevators on a clockwise ring? The first placement is arbitrary. It just sets the origin. Then there are six possibilities of what comes next, then five possibilities of what comes next, and so on, and the answer is six factorial, 6 x 5 x 4 x 3 x 2."
"But any ring sequence and the associated rings of the larger-size jump codes, two to n minus one, form complete ring sets in the mathematical space. Each ring in the set of six has the same topology as the five others. Any of the six rings can act as the base ring for the other five. So knowing one complete path divides the six factorial option space by five factorial. There are only six rings left."
Mark shared another kiss with Emily. "And that's what happened in Aggie's carriage zone puzzle. This is a great example of how a little thinking can save you a huge amount of brute-force searching. The original number of possibilities for the ordering a ring of 673 elevators was huge, 762 factorial, but the complete path on one tumbler code reduced the original possibilities by 761 factorial, leaving a single set of 762 rings. There are two base rings in this set where the entrance and exit nodes will indeed be adjacent, a clockwise and counter-clockwise pair. But it doesn't matter. All these rings are topologically equivalent. My breakthrough came when… well, when I realized this."
Fatima nodded. "Got it. Thank you for a tour of your mind." She paused and then changed topics. "Where do you think this pyramid is?"
"I've been worrying about that too," whispered Emily. "Four meters by 36 floors, we'd see it on the skyline if it were anywhere on the surface around here. Plus there's the fact of the dark night and the snowstorm. What do you think Mark?"
"Well, the pyramid's footprint would be a little too large for an enclosure at Black Mall's diamond complex, unless maybe one enclosure goes much further down and then widens at the base. Remember the diamond complex is not over Wobanakik. There's physical room for it."
"Or the pyramid could be contained in White Mall's diamond complex," countered Emily. "We've never found a way inside. It seems to be one big structure, more than enough room." She felt Mark shudder in her arms.
"Oh Aggie," Mark whispered. "Come home safe! I tried to keep the worry out of my voice when I was talking with her, but I really want her back home. Given what we know about jump technology, she might even be on a different planet now."
Chapter 84.
One hour later.
Time: Thursday, May 14, 2019 3:49 PM
Tom and Carla were riding stair buggies around the second floor perimeter of the White Mall diamond complex, finishing their scheduled afternoon mission of supplying the Wobanakik entrance rooms. They had their five-speed buggies set to the middle gear, three meters per second, and the journey from the top of the pit of the great stairwell to the White Mall lounge would take fifteen minutes. Behind their buggies four more unmanned buggies were obediently following them.
After their April escape from Wobanakik, Tom and Carla looked for multiple ways to be useful in their new society, and one of the things they did was to spend their evening hours together at the Hilton, using the mall remote monitoring system and diligently probing the many stores for hidden treasure. The last day in April, four days before they would leave for their eight-day stay at the Bee Park, they make a great discovery.
Their target was Speedo's Stair Assist, a small 10-meter-long store situated near the 22 km spiral mark of Blue Mall. The quickest way to reach it was from the Blue Mall sky bridge, and after a bit of puzzle time at the store, they realized they had found something fabulous, six programmable carts that could climb stairs.
The carts were a marvel and used a technology neither Tom nor Carla thought came from Earth. A dense array of almost a million tiny rods formed the bottom of each cart, and the rods would flow up and down and conform to the contour of the terrain underneath. What powered the horizontal motion was unclear.
With a maximum capacity of 150 kg and maximum cruise speed of 5 m/s, the buggies were ideal for supplying the Wobanakik entrance rooms. In their middle gear, they would climb or descend the great stairwell to Wobanakik in less than nineteen minutes. Riders were still needed of course. The buggies could not manage the passage through the bottom elevator, but with just a little help the set of six allowed two people to deliver 600 kg of cargo to the entrance rooms in about ninety minutes round trip as measured from the White Mall lounge, and that included over twenty minutes to unload once the buggies were on the other side of the entrance elevator. Sandra and her folks below were still singing praises about Tom and Carla whenever they were on the network.
The buggy rides were interesting, super smooth with linear travel but with sharp accelerations in the curves. It took a little getting used to, riding the buggies at their top speed, and unless they were in a hurry, Tom and Carla limited their velocity to 3 m/s. They had popped out of the great stairwell five minutes ago, and after a quick climb to the second level began the final leg of their journey, orbiting the perimeter of the White Mall diamond complex.
"Ten minutes Suvarna," Tom called out into his Leophone. "We're right on schedule." He, Carla and Suvarna were on a private line.
"Hi Tom," Suvarna replied. "I'm in the kitchen starting dinner. Be nice to have you back. Transfer go okay?"
"Yep. Everything's fine. Our four empty chicks are tagging right behind us."
"Tom!" Carla called out behind him. "Did you just call me a duck?"
"I'll get back to cooking," laughed Suvarna. "See you in a bit." She would be opening the vestibule door for them when they arrived.
Tom stared at the outside scenery for a moment. The past two months of spring had been lovely weather, with warm days now but the short nights were still delightfully cool. Tom thought it was a fabulously nice climate. The only unknown now was how hot the summers would get. The summer solstice was fifteen days away, and at their latitude the sun would not set on that day.
Tom called out to Carla behind him, "You've had a busy day today. Want a massage tonight?"
"A back rub would be nice, yeah. Thanks." Carla paused for a second. "It really wasn't a hard day physically, just a stressful one. The walk to Green Mall's trapdoor was a stroll in the spring sunshine, and riding these carts is like being on an amusement ride. I just wish I could have talked Fatima into letting me go with Aggie. It's not fair for Aggie to be facing this danger alone."
"Yeah, I know. I understand Fatima's point, but yeah…" Tom took a deep breath. "On the call today, once we have a clear understanding of how to get back, Fatima seemed receptive to Mark's idea of returning to the pyramid in force."
"God, I hope so."
"I'd like to go with you Carla."
"Really? Thanks Tom, much appreciated. I can't think of anybody I'd rather have by my side." Carla leaned back in her buggy and sighed, hanging on just a bit as her cart made a sharp sixty-degree turn to travel the next 100-meter hexagonal segment.
After the turn, Tom punched his phone to mute and then switched lines to listen to Aggie's progress. Her voice came through loud and clear every six seconds. "V-17, 254, wall… V-18, 197, door… V-19, 174, wall…"
Aggie's routine had evolved to optimum efficiency over the last hour, supported now by everyone at the Hilton. Whitney and Amber were logging Aggie's report and verifying each other's work. Akiko, Emily, and Alison were running the reverse lookups on the base ring and computing the tumbler jump sizes. They were currently running about fifty jumps behind Aggie, but the lookup delay didn't matter, just the quality of their work. And finally, Heather and Ashley were looking at the data of what Aggie was seeing as the elevator opened.
No wonder she hadn't found anybody yet. The group's model for the transfer rooms was that all 91 carriage rooms on the 18th floor and above were in perimeter rooms with the carriages facing the transparent walls. For the 19th floor and below, the other 582 carriages were always NOT in a perimeter room, and five times out of six the carriage would face a blank wall. Carriage orientation seemed random within the rooms and all the transfer rooms that Aggie had investigated had one exit.
Five times out of six, that exit was a closed door, and the typical Kappa Alpha door was absolutely soundproof. Even in the rare times Aggie's door opened to an arch, she thought she had a small chance of being heard. Kappa Alpha environments had a strong tendency to deaden sounds from other rooms.
Aggie found all this information encouraging. Her missing Party was here! She knew it. They were just going to be hard to find, that's all. Aggie was optimistic. Aggie was psyched. Aggie also had to go the bathroom. "V-23, 169, wall… V-24, 146, wall…"
After each jump she was calling out the new tumbler code for her current jump, the physical carriage of where the new jump had carried her, and what she saw through the open portal. "V-25, 60, transparent perimeter wall." She had trained herself to always click the new code before pushing the close button on the elevator. "V-26, 133, wall." Aggie had learned that lesson the first hour she was here. "V-27, 565, wall." Pushing the elevator close button locked the tumbler code. "V-28, 5, transparent wall and stop."
There were a few seconds of silence on the bridge. "And stop?" asked Amber. "You see something?"
"Just the usual darkness and swirling snowflakes," Aggie replied with a yawn. "Wow. I feel a bit tired. I also need to find a restroom."
"That's a great idea," Akiko said. "Take your time Aggie. It'll give us a chance to catch up with the jump-size computations."
Another yawn from Aggie. "Yeah. Thanks Akiko."
Emily spoke a gentle warning. "Aggie, you're on the last floor with one zone and carriage. Floor six has two zones. Floor 5 should have 61 rooms. That's enough to get lost in, especially if you lose your orientation."
"Yeah, that's for the warning. But I really need to find a bathroom. I'll try to keep my motion linear." But that proved to be difficult. The openings in the honeycomb of rooms did not align. Every room she visited had doors on three of the six walls, but the doors were never opposite each other. After sixteen rooms (not counting the transfer room), Aggie considered just relieving herself on the floor and letting the servos handle her mess, but then the seventeenth hexagon turned out to be a bathroom. Aggie had to laugh at what she saw inside.
In some sense, the room didn't have an inside. It was an extremely open bathroom even by the typical Kappa Alpha lack of concern for privacy. It was a very functional bathroom to be sure, a few toilets and sinks and a shower area designed for two people. But it was also a perimeter room, and on a corner of the collar ring. The room had with three exterior walls and a transparent ceiling. And the modesty walls of the toilets were insanely short. They formed a short skirt that shielded her hips as she sat on the toilet and little else. Aggie could easily have an eye-to-eye link or a verbal conversation with a person in the next stall if it were occupied. And people in the perimeter rooms above on Floor 4 would have a great view of her anywhere in the bathroom.
Aggie turned around and gazed at the rooms above. They were dark, which was typical of perimeter rooms but not interior rooms. On the perimeter, low level lighting would activate only when she was about to enter the room. It was adequate to see everything but noticeably dimmer than the bathrooms at the home complexes. There was compensation though for the low lighting. It made the snowstorm very easy to see.
Aggie didn't bother to close the short door on her stall. She just plunked her two backpacks down outside and breathed a big sigh of relief as she drained her bladder. She sat for a while, her pants at her ankles, and just watched the snowstorm for a while. The night wind was very brisk and the swirling snow soothing to watch. Aggie felt herself relaxing and decided to do a bowel movement too. Perhaps the Kappa Alpha builders weren't so insane after all with their glass bathroom. It felt kind of nice, emptying her bowels in a clean, quiet, and dimly lit bathroom, warm and dry and surrounded by blowing snow. Aggie felt her mind was at peace when she finally stood up and cleaned her bottom.
She shook her two canteens and decided she would refill both of them before leaving the room. She was then a bit startled when Emily's voice came from her Leophone. "We're all caught up now Aggie, and we also followed your lead and did our restroom breaks. We're ready to resume when you are. If you're curious, the jump size for V-28 was 434 on the base ring."
"Ah." Aggie sighed as she switched mental gears and prepared to get back to work. Pants up and zipped, all that was left was to get some water from the sink and she would be ready to go. "How long have I been on break?" she asked.
"About ten minutes. There's no rush. You want to rest for a while?"
"No, I'm okay. The night scene outside is lovely. It's making me a bit sleepy. I think I'll sleep very well tonight."
"Yeah, I can imagine," Emily replied. "Find your bathroom okay?"
"Yes. It's on the perimeter, very open. I didn't like it at first, but I do now." Aggie laughed. "I'm being serious! Taking a crap on a warm toilet in the middle of a snowstorm is kind of nice!"
Emily laughed back. "I'll take your word for it." After a short pause she added, "Was it tricky to find the bathroom?"
"No, just a bit long, seventeen rooms from the carriage I think."
"You think? You didn't count?"
"Well, I always chose the door on my left when I had a room with more than two doors in it. Now I'll just walk back and choose the doors on my right." After filling her canteens, Aggie returned directly to the transfer room, not bothering to explore the silent zone of rooms. That work would come later.
A few minutes later she said, "Okay, I'm back at the carriage. Hey Mark! You still around?"
A few seconds of silence followed, and then, "Oh, I'm here. I had myself on mute."
"Mark, since I started jumping the different tumbler codes, this is the third time I've hit carriage 5. There were other numbers I'm hitting a lot too. I'm sure of it. Is that what you would expect?"
"That sounds completely reasonable. For how we're doing the jumps now, we're not going to hit every zone. I would guess we'll miss, I don't know, a third to a half of them maybe, miss less than a half I think. And that means we'll hit other carriages multiple times. I'll pull up the complete distribution for you after the X-28 jump. Hey Aggie?"
"Yeah?"
"I just remembered something. The Leophones are programmable. Once you finish with X-28, I can download the base ring and the complete tumbler jump vector into your phones. It's a trivial amount of data and it'll just take a moment to translate the programs to run on the phone. Then you can run the jump program locally if you need to. Just type in the origination and destination carriage numbers, and the phone will display the appropriate tumbler code."
"Wow, that's great! Thanks Mark."
"Yeah, just as a backup. It'll be good insurance. If you want to test a ring, whatever carriage you're in, just set the destination to the top of the pyramid and if you get there, your tumbler code is not the broken ring."
"Yeah, I get it. And it's time to get back to work. Tumblers clicked to W-1 and closing from zone 5… New physical carriage is 248 and I see a wall… W-2, 172, door… W-3, 615, wall..." Aggie could feel the eagerness rising in her as she worked. "W-4, 467, wall…" The W and the X series were the final two sets of codes. "W-5, 193, wall…" Would she really be home by 7 o'clock tonight? "W-6, 506, wall…" Arriving to a sun-filled late-spring evening would feel so strange after spending so much time in a dark night's snowstorm. "W-7, 259, door…"
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