It's My Party

"I'll give you an example. Imagine six codes and a set of elevators numbered one to seven. Tumbler code 1 gives you the ring 1-2-3-4-5-6-7-1, code 2 gives 1-3-5-7-2-4-6-1, and code 3 gives 1-4-7-3-6-2-5-1. Code 4 means four jumps up which in modulo 7 is the same as three jumps backwards. Code 4 is code 3 in the opposite direction. Jumping code 4 means going from 1 to 5 to 2 to 6 and so on. Code 5 is code 2 in the opposite direction, and code 6 is the opposite of code 1, counting back by 1. Because seven is a prime number, the six rings never collide."

"But how do you get out?" asked Madison.

"Because there's an exception," said Aggie. "Exactly one of the mathematical rings is broken. Using Emily's example, imagine the broken ring is code 2. Instead of 1-3-5-7-2-4-6-1, the actual jump pattern is 1-3-5-7-2-4-0 and 6-1, where zero is the exit carriage outside the zone. Code 2 in elevator 4 jumps me out."

"And you don't know to where?" asked Fatima.

"No," replied Aggie. "But here's the good part about all this. The missing people from Party #4 would not know how or why to search for the broken-ring. I'm almost certain they are still in this pyramid, and I'm very hopeful they're maintaining a presence in the transfer room of their zone. All I have to do is stick with a fixed tumbler code and start jumping. I should find them in less than an hour."

Fatima blinked. "That fast?!"

"Yeah, I think so, unless I'm very unlucky and leave here because I picked the tumbler code for the broken ring. There's a very small chance of that happening, one in 672, and even if I do, I have a 50% chance of finding them before I jump the broken link in the ring."

Aggie paused to take a deep breath and then continued. "The elevator door travels at one meter per second, three seconds to open, three seconds to close. I can push the close button as the door starts to open and grab the door if I change my mind. Otherwise I'll jump every six seconds. At ten jumps a minutes, I'm estimating 68 minutes for 673 jumps. I'm assuming I can exhaustively test eight tumbler codes per day. If I'm doing this solo, I estimate twelve weeks to test everything, six weeks with average luck to find the broken ring."

"Aggie," said Emily. "I mentioned earlier a nice factor of two. Remember how in my example you picked the broken ring to be 1-3-5-7-2-4-0 and 6-1? That means carriage 6 can only be reached by five codes, and not from carriage 4. All the other carriages can be reached by six codes."

"Uh, yeah, that right."

"Aggie! You know where carriage 6 is! It's the entrance carriage at the top of your pyramid! Don't you remember the lesson? On the broken ring, the entrance into the zone and the exit out of the zone are a matched pair of carriages."

"Yes, I remember now! I just never realized the implications. The entrance conduit fills the other half of the broken link. The broken ring will not jump to the entrance carriage. So if I'm testing a tumbler code and I arrive at the top floor, I know my tumbler code is not the broken ring."

Emily nodded. "That's right. So on average, you'll be testing 50% of each tumbler code before you arrive at the top floor and realize you're on an unbroken ring. What you should do then is make one more jump with the old code and then pick a new ring code to test."

Aggie smiled as she sat in her lounge chair. "Right, make one more jump, because otherwise if I pick a new tumbler code from the top carriage, I'm guaranteed to make 673 jumps to either to return to the top carriage or exit the system on my last jump. I lose my nice factor of two if I start my search from the top carriage. So, 34 minutes of searching per ring, maybe 16 rings per day, yeah, in three weeks, I can search 336 of the 673 rings. Yeah… Fatima, this really isn't so bad. If all eighteen people here were searching in the right way, it would only take us a couple of days. We could divide the tumbler codes into segments."

Emily added a suggestion. "Maybe give everybody a unique position on the first tumbler to start things off." She then switched topics. "Aggie, I should tell you the results of our Plan-C testing."

"Yeah. I was wondering why I didn't see Cassidy's red letter."

Emily then went on to explain the results of their testing with Cassidy's Leophone.

"Interesting," commented Aggie when Emily was finished. "I wonder if my exit link will take me to where the missing phones are. There'd be a certain elegance to the topology if it did."

Cassidy spoke up. "Aggie, I'm very prepared and willing to hike back to the trapdoor and join you. I can help you search too. Just tell me how to find you after I get to the top of the pyramid."

"Uh, I appreciate that Cassie, but why don't you hold up a bit? The only way to send goods through to here is for a person to come with them. I might decide I need something. You can bring it with you later."

Fatima felt torn. Aggie's point was excellent, but still… "Aggie, are you really comfortable about exploring the pyramid alone?"

"I am Fatima. And I can see my second Leophone has just activated. I'm going to start my search now. I'll keep both lines open. Hopefully I'll find Party #4 within the hour. If I make a full loop and find nothing, maybe Cassidy could join me then." Fatima agreed, and with numerous people cheering her on, Aggie got up and headed for the carriage numbered 12:28.

Chapter 83.

Status check: Current position (and original Party affiliation) of the human species (six men and seventy-six women).

Manhattan, Wobanakik: Jada(1), Thara(2), Oona(2), Abit(2), Lynn(5), Mandy(5), Charles(5) (All unboosted)

Entrance Rooms, Wobanakik: Madison(1), Sandra(2), Amy(2), Margaret(2), Sachi(2), Parni(2) (All unboosted)

Bee Park, an environmental enclosure inside the Black Mall diamond complex: Three members of senior Boost Team 4, arrived at Bee Park May 8, 2019: Fatima(1), Husna(1), Kiyoko(5). Three members of junior Boost Team 5 and certified medical technicians, arrived at Bee Park May 12, 2019: Lucia(1), Tajana(2), and Paige(5). Graduate of original Boost Team Zero: Mark(1), first arrived at Bee Park January 17, 2019.

At the White Mall home library, in charge of supplying the Wobanakik entrance rooms, the three graduates of Boost Team 3: Suvarna(2), Carla(2), Tom(5)

At the Green Mall home library, having returned from their mission to the Green Mall trapdoor: Two graduates of Boost Team 2: Ann(5), Cassidy(5). Graduate of Boost Team 1: Hannah(1). Also present: Graduate of Boost Team Zero: Frida(5)

Wandering the hexagonal pyramid: The assumed 17 lost members of Party #4, plus Graduate of Boost Team 2: Aggie(4).

The seven lost members of Party #6 (Yellow Mall): location(s) unknown

The seventeen lost members of Party #3: (Blue Mall): location(s) unknown. The combined lost parties of Blue and Yellow Malls consist of one man and twenty-three women.

The three additional members of Party #5, currently in the Black Mall surface area: Jessica, Ricardo, Diego.

At the Black Mall home library, on duty as backup for the Boost Teams: Boost Team Zero graduate Toshi(5), Boost Team 1 graduate Holly(5)

At the Hilton Conference Center, the terminal store of the Green Mall spiral: Boost Team 1 graduate Emily(1), and the remaining six (unboosted) members of Society: Ashley(1), Akiko(1), Alison(5), Heather(5), Whitney(5), Amber(5)

Time: Thursday, May 14, 2019 1:18 PM

Everyone in the Society was on the network and listening to Aggie's progress. They were waiting for her to report she was back in her zone's transfer room on floor 12. At the Bee Park, Fatima suddenly felt Mark stiffen beside her. "You okay?" she asked quietly.

"Holy, holy shit!" Mark whispered in reply. Fatima turned and looked at him. He was grinning broadly and he stood up. "Fatima!" he announced. "I need to get to a computer! I need a computer right now!"

Fatima nodded and said playfully. "Well, go get one tiger!" She decided not to bother to ask for an explanation. Mark seemed so desperately eager to go.

"We'll be here for you Mark," Holly confirmed over the bridge.

It would be no problem for Mark to get back to the Black Mall library. There was a mushroom lock at the top of the towering stairwell that would allow him to exit the Bee Park enclosure, and later the door out of the diamond complex was exit-only which was the direction he was going. The only assistance he would need would be for Toshi or Holly to open the vestibule door and let him into Black Mall's home complex. Mark would simply ask for help with his Leophone for that.

He stayed on the phone as he ran across the park. "Aggie, you haven't started jumping yet, have you?"

"No, I took a small detour. I was passing by a kitchen. It's very encouraging. It's full of food and there are also small amounts missing. I couldn't ask for better."

"Uh huh…" Less than a minute later Mark was out of the Bee Park simulation and dashing across the short series of entrance rooms on his way to the staircase. Meanwhile Aggie completed her walk to the transfer room. She opened the elevator door and entered the carriage. Her search for the lost members of Party #4 was about to begin.

"Hi Mark. I'm just about to start. I'm staring at the tumblers now."

"The jump codes, the 672 possibilities, are they digital?" he huffed back.

"Sort of. They're discrete, but it's not a digital display. The tumblers are analog-style dials, what you would find on the face of an old safe. The left dial uses the letters A through X, and the right dial uses the numbers 1 through 28. Both dials have a bright chrome finish with sharp engraved black lettering, and there's a raised knob in the middle for turning the tumbler. They feel very-well machined, typical Kappa Alpha technology. They spin silently, but there's a definite tactile click when I change codes. I can feel it."

By this time Mark was starting his race up the 1296 stairs to the surface. "Great," he replied to Aggie's description.

Aggie could sense something was up. "You've got an idea?"

"Yeah. This is so cool!"

"Any particular tumbler pattern you want me to try?"

"Uh… No, pick anything," Mark said in a gasp as he ran up the stairs two at a time. Then he slowed down a bit, deciding he had to pace himself. "It's not the tumbler pattern. It's the order of the zones that's important. Emily! Aggie! Aggie, can you call out the carriage numbers as the doors open? And Emily, can you record them? The floor numbers too if you want Aggie, but it's the carriage numbers that are critical."

"Sure Mark. I can do that," replied Aggie.

"Sounds like a very good idea," added Emily.

"It's an absolutely brilliant idea," Mark countered joyously. He laughed as he ran. "Oh, this is going to be so neat!"

"You got it," said Aggie. "I've set the tumblers to A-1, pressing the close button now from carriage 28, floor 12." A few seconds later she added, "I'm on the bottom floor again, floor 36. The carriage number is 635. I'm facing a blank wall, poking my head out. Hello?" she called out in a loud voice. A few seconds of silence went by. "I didn't get any answer. Mark? I don't see a doorway on my three sides of the transfer room. Do you think I should walk around, maybe shout through a doorway?"

"One second…" He slowed down his run so he could talk more easily. "No. Actually I'd like to suggest we postpone the search for Party #4. Aggie, will you just keep jumping the A-1 code? I mean all the way back to carriage 28. Just keep calling out the carriage numbers. The order will be fabulously useful."

"Yeah? Okay, sure. This will take me sixty-eight minutes if I speed jump and do nothing else. I'll stop of course if I see people."

"Of course. This is great Aggie, thanks. I'll be in the Black Mall library long before you finish the full ring. Emily, pipe me the data as it comes in, okay?"

"It'll be waiting for you," she replied.

"Thanks!"

"Here I go then," added Aggie. "I'm pressing the jump button now… New carriage is 208, floor 25… New carriage is 245, floor 26… New carriage is 98, floor 19… New carriage is 536, floor 34… Carriage 347, floor 29…" Every six seconds, Aggie was calling out a new report. "Carriage 600, floor 35…"

Seventy minutes later.

Time: Thursday, May 14, 2019 2:31 PM

Aggie was counting off the few remaining jumps on her unbroken ring. "Carriage 407, floor 31." The last hour had been very discouraging. "Carriage 507, floor 33." No trace of the missing members of Party #4 had been found. "Carriage 346, floor 29." And the views out of the elevator in all the different transfer rooms had revealed very little. "Carriage 239, floor 26". On the top half of the floors, transfer rooms were always in perimeter rooms. "Carriage 150, floor 22." The views always faced outward, showing the dark snowstorm. "Carriage 348, floor 29." And Aggie at this very moment in carriage 348 was exactly one hundred jumps from the entrance room at the top of the pyramid. "Carriage 340, still floor 29." She knew tumbler code A-1 was not the exit code. "Carriage 228, floor 25." And as for the transfer rooms below floor 18, they were almost always showing her the blank wall of an interior transfer room. "Carriage 218, floor 25." Aggie would have had to walk around to the opposite side of the room to see the exit. "Carriage 418, floor 31."

"One more jump to go," said Emily in a gentle voice.

"Tranquility base here," Aggie replied, trying to crack a weak joke. "The eagle has landed at carriage 28, floor 12. Happy Mark?"

"Yes, happy Mark?" added Emily from the Hilton. "I've sent you all the data."

He replied in a very friendly tone of voice from the Black Mall library. "Extremely happy. Emily, I'm sending you some computer code in return, and Aggie, you have just given me the key that unlocks the pyramid."

"I'm very glad to hear it," she replied. "I'm also glad one of us is happy. I'm feeling a bit discouraged."

"Ah, sweet Aggie, I think I can cheer you up. Ready for a few more jumps?"

"Sure. Want me to pick a new tumbler code?"

"Yes please. It doesn't matter much which one. Just pick a progression that will be easy for you to follow."

"You got it. Tumblers are now set to A-2."

"An excellent choice. You may fire when ready."

"Right," Aggie commented dryly. A few seconds passed. "I'm now at carriage zone 154, floor 22." A moment passed. "Want me to jump again?"

"Not right now. Just hold your position for a moment… Aggie, Emily, I've laid out the complete path of the journey on tumbler code A-1 on a circular ring. I'm calling that the base ring, with its origin node at carriage 1. Code A-1 makes unit jumps along the base ring, node to adjacent node, that's the way I'm defining the base ring. Your previous carriage Aggie, number 28, was 105 jumps from the origin node on the base ring."

"Okay," replied Aggie. "I'm following this."

"Now, here's the really good part!" said Mark laughing. Your current carriage zone of 154 is at position 608 on the base ring. You did that in one jump, moving from position 105 on the base ring to position 608 on the base ring. Therefore tumbler code A-2 had a jump size on the base ring of plus 503."

There was stone silence on the bridge for a moment, and then Emily called out loudly from the Hilton. "My God, my God! I feel like such an idiot!"

Mark continued. "The next jump of 503 steps will put you at position 438 on the base ring, which corresponds to physical carriage 287. Try it Aggie."

A few seconds later, "It's just as you said. I'm at carriage 287, floor 27."

Emily interjected. "Mark, I want you to know I bear you no ill will for showing that I belong in mathematical kindergarten."

Aggie spoke up from her carriage. "Emily, if this is putting you back in kindergarten, where is it leaving me? I still don't get it."

"People around me, beware!" said Mark laughing. "I'm going to be insufferably pleased with myself for at least the next few hours!" He then got serious. "Aggie, I know your entire path on tumbler code A-2. You're actually not that far from testing whether A-2 is the code for the broken ring. If you jump that code 57 times from your current position, you'll be at position 170 on the base ring which is physical carriage number 615 on the 35th floor. The next A-2 jump will be crucial, because 170 + 503 is 673 which means you will be jumping to the origin of the base ring. Get it? You'll either be at the top of the pyramid or out of it."

"Oh my gosh! Yes, now I get it. Emily, mathematical nursery school for me! Mark, you can get me to position 170 on the base ring a lot faster than 57 jumps, can't you?"

"Yep!" A few seconds of silence passed by. "Uh, Aggie, maybe not. Our known jump sizes on the base ring are 1 and 503, and we're not at a good spot to get to base ring position 170 quickly. Four jumps forward with 503 is the same as seven unit jumps backwards. Without lots of jumps, I'm locked into four regions of the base ring, and none of them is near the region below 170. But the important thing is we know that the A-2 jump size is 503, and we can test A-2 later when we can get to base ring position 170 easily. What we need now are more options for our jump size. So pick and jump another code."

"Okay! Tumblers set for A-3. Jumping now… I'm at carriage 137, floor 21."

A few seconds passed and then Mark said, "Which is position 572 on the base ring, and you moved from ring position 438, so the base ring jump size for A-3 is 134… Hey, really?!"

Aggie replied with a laugh. "Mark, would I lie about a thing like this?"

"No, I mean this is great! Aggie, jump code A-3 two more times. That will get you to physical carriage 547, floor 34."

A moment later Aggie replied. "Done. I'm here."

"Aggie, you're also at base ring position 167. Set jump code to A-1 and"

"And jump three times! I get it!" A half a minute passed before Aggie added, "Resetting tumblers now to A-2 and jumping again… And it worked! I'm at the top of the pyramid! So code A-2 is not the broken ring! Hey Mark, this is great! So big guy, what's next?"

"Uh…"

Emily spoke up from the Hilton. "Mark, if I may be allowed to knock you off your high horse just a little, the right way to proceed is to postpone testing for broken rings until we can proceed more efficiently. We should concentrate on learning the jump sizes for the tumbler codes. I meanall of them."

Aggie answered first. "Yes, I see it. It's obvious. I'll click a code and jump, click to the next code and jump. I'll tell you the physical carriage number, and you guys can convert it to movement on the base ring. I can probably do this almost as fast as pure speed jumping with the same code."

"I see it too," added Mark. "Great thinking Emily. Everybody, Aggie is going to find the complete mapping of tumbler jump codes to step size on the base ring. Then we can rapidly test the rings. Imagine two jump codes alpha and beta whose sum of jump sizes on the base ring add to 673. From the top of the pyramid, you jump with code alpha and then with code beta. You're either back at the top of the pyramid or out on the broken link. If you're back at the top, you then jump first with code beta and then with alpha. You will have tested two codes with four jumps. What do you think Aggie? Thirty seconds?"

"Yeah, good guess."

"And there are a total of 336 pairs of codes. And once we have the jump code mapping, we start eliminating them at two pairs a minute, so figure three hours, and with another seventy minutes to get the jump map…" Mark looked at the time. "Aggie, we can get you out of there by 7 PM tonight, even if the broken ring is on the very last remaining tumbler code."