Verdania: A Mathematical Odyssey

Ok.

I have created yet another blog for my POW stories:

http://verdaniaamathematicalodyssey.blogspot.com/

Verdania is an island, somewhere in the Carribbean (read Bermuda's Triangle, perhaps) where a crew of people (Gilligan's Island or Lost?) are shipwrecked. They find an indigenous culture in constant war with a mysterious enemy. They also find other mysteries along the way.

The story is in constant evolution and I want to collaborate with other people on this.

Please take a look:

Heck, you might even like them enough to help me move forward with the story as well as the math.

Thanks

Chapter 10: Revealing

It was a vexing dilemma that faced the shipwrecked crew on this island they came to know as Verdania. They couldn’t get enough information to understand where they were, much less how to leave. They were living in a tree house village of young people with abundant food and a tranquil lifestyle. But they sensed some latent danger lurking.

It was Miguel who finally took action. “Look, we accomplish nothing by staying here and waiting,” he said quietly in the company of the Captain, Cynthina, Jeanie and Marge. Justin and Jake were off with Teena and the other kids. “While they are nice enough, it’s obvious that we don’t fit in. They don’t really know how to help us anyway. I say we offer to accompany them on the next trip they take to deliver those mangos. I would bet that the adults will be a little more helpful.”

Teena and Justin came up to them. They explained their idea to her and she agreed whole-heartedly. “But you know, Justin and Jake really should stay here with us. Youngsters don’t belong on the Rim.” This puzzled Miguel and caused a worried Cynthia, Justin’s mother to react.

“I can’t just leave my son behind!” She looked at the Captain with slight panic. 

The Captain nodded, then said, “Ok, this is what we’ll do. Miguel, go with Jesse, Jeanie and Marge to the Rim and find out all you can about this island and its inhabitants. Cynthia and I will stay back here with the boys and explore this valley. I’m betting there is a way to send a message back to us.” She looked at Teena but saw absolutely no reaction. Was she even listening?

The next day the four college students took off with the two oldest members of the village to deliver the mangos. It was a long and tedious journey because they had to leave caches of mangos along the route. Jeanie and Marge were sure there was a better way to do this, but it was just another mystery amongst many. It bothered Jeanie to no end, but amused Marge, who found these customs charming.

The entire trip, after all the backtracking, took them almost a week. They had brought their own food so as to not disrupt the mango delivery system (a system, according to Jeanie, that was pointless and inefficient). The valley floor was flat and the path they took perfectly worn with the three grooves for the cart. About 70 kilometers into the trip they passed a perfectly round and placid lake with huge orange fish swimming near the surface.

When they were finally approaching the Rim, a band of adults was waiting for them by a huge stone monolith alongside a beautiful lake. Two women came running forth and embraced the two young villagers who had accompanied them. The villagers looked at each other, confused. They were swept away by the two excited women who were chatting as if they had always known these two.  The other adults gathered around the Miguel and crew. They looked happy and welcoming, sort of like Teena and Otto had when they had first arrived in their village, but quite a bit more interested in the details.

“Welcome to Verdania. We have heard of your rather dramatic entrance onto our island. I am Dula.” Dula explained that she was the leader of the group of adults living on the Rim of the Great Verdania Caldera Valley. 

“You know about us?” asked Jeanie, incredulous.

“Of course we do. Nothing enters or leaves Verdania without us knowing. This is a dangerous island, as your boy Jake would have found out if we hadn’t saved him from the Deesors.”

“Deesors?” 

“We have a lot to explain, I know. But first we need to get back to  Sentry Point One. Follow me.” They started up a steep, serpentine path along the cliff wall that enclosed the valley. The famous Rim was like the edge of  a huge saucer. As they climbed up the trail, the valley came into spectacular view. Dula walked effortless up the path, all the while explaining the layout of the Verdania Caldera Valley to the new arrivals.

“The circular Caldera Valley has a 100 km radius. The Youngsters’ village is exactly in the center.  Here at Sentry Point One we are at the northern most point of the Caldera Rim. Hope Lake, which you passed on your journey, is located due south 30 km from here. Starting at Hope Lake and going due west you would eventually reach the Sentry Point Two on western portion of the Rim. At that point, if you go due south for 15 km, you would reach the Adult Village.”

Jesse was perplexed. It seemed to him that just like the Youngsters, the Adults were going about their tasks in confusing and inefficient ways. “Why not just make a path along the circumference? Wouldn’t it be shorter?”

Dula looked at him equally perplexed. “My dear man, our route IS the shortest route. And besides, it’s far too dangerous to be moving along the Rim on a regular basis.” Just then a huge rock came flying through the air and hit Jesse on the side of the head. He crumbled to the ground. Danger was apparently an understatement on the Rim, thought Marge.

Your Task:  “As the crow flies”, just how far away is the Adult Village from Sentry Point One? How long would the route be if it included a walk along the Rim? How long is the route the Adults currently use?

Chapter 9: Harvesting

Teena and Oto, the leaders of a very young tribe of tree dwellers, proved to be excellent hosts. They had saved Jake from some life threatening attack (which he still did not understand) and provided food and shelter for his Aunt and the other shipwrecked passengers once they had calmed down and understood that Jake had not been kidnapped in reality. Justin was instrumental in convincing the Captain, as well has his mother, that they were not in any danger. It wasn’t that he knew this for a fact. It was more of an intuitive feeling. Something in the serene and joyful voice of Teena inspired confidence. He felt safe.

For her part, Teena seemed to be drawn to Justin and Jake as well. She showed them around the village, which was built inside a grove of huge mango trees. The villagers lived in huts built high in the branches and accessed by rope ladders hanging down during the day and drawn up at night. Between the huts was a system of rope bridges that connected them together, resembling a horizontal spider web of sorts. Justin loved walking back and forth across the bridges. The mango laden branches hung down as if offering up their fruit to the villagers. The villagers harvested the mangoes and placed the extra ones in a bin near a cart at the edge of the grove.

Although the Captain could sense there was little immediate danger, she could not ignore a nagging feeling of discomfort at the way Teena ignored her, the obvious leader of this group, in favor of Justin and her nephew, Jake. She found it odd that the oldest people in this tribe were in their early 20’s. They all seemed quite competent and even skillful in the way they took care of their village, but it was strange none the less.

The Captain tried to catch Teena’s attention several times to ask where the adults were. She had told Jake that she and Oto were the leaders of the tribe. But where were the parents, the younger kids and the grandparents? She couldn’t get a straight answer from Teena. Finally, she confided in Cynthia, Justin’s mother, about her worries that some terrible information was being withheld from them.

“You know something? I’ve been wondering the same,” said Cynthia quietly one day. “It’s like they are orphans or something. They are polite to me, to be sure, but they rarely talk to me about anything real. It’s almost as if they look straight through me when I am with them. I feel awkward asking them again.”

“I’ve been thinking, why don’t we ask Justin and Jake? They seem to have an easier time talking with Teena.”

The boys were confused by the Captain’s request when she spoke with them later that day. They hadn’t noticed that Teena and Oto were basically ignoring the adults in favor of them. But now that they thought about it, they also began wondering about the adult villagers. They asked Teena about them.

“Oh, Adults are far, far away from here.”

“Why?”

“Why? Because that is the way of things. Adults live on the Rim and Youngsters live in the Grove. “

“Don’t you miss them?” asked Justin.

“Miss them? Why? We send them mangoes each month.”

Jake and Justin looked at each other.

“Adults are very busy people. We send them mangoes to help out and let them know we are ok.”

“How far away is the Rim?”

“It’s about 100 kilometers away. We have one cart and it holds exactly 100 mangoes. When the Messengers are sent off with the mangoes, they have strict orders not to eat more than one mango per kilometer between the two of them. If they ate any less, though, they would die.” 

Justin did a quick mental calculation and blurted, “Don’t they end up eating all the mangoes before arriving?”

“Oh, That is why we wait until we have 300 mangoes ready. We’ve developed a system of caches, sort of like holding bins, to leave mangoes along the way.  The Messengers go back and forth between the Grove and the different drop off points. At some point, they have collected all the mangoes they can at a certain drop off point and they set off to the Rim with their final load.”

“I’m confused,” said Jake. “How do the Messengers get back? 

Teena looked at him placidly, “Come back? Why would they do that?”

Your Task:  Determine the maximum number of mangoes the Youngsters can get to the Adults on the Rim?

Chapter 8: Riddling

Jake awoke to find himself swaying in a hammock. He opened his eyes with a jolt, then closed them quickly when he felt as if his head were exploding with pain. He reached up to his temple and found it bandaged. He slowly re-opened his eyes to look around. This wasn’t some movie where he might awaken, disoriented and thinking it had all been a terrible dream. No, he knew he had been kidnapped. And he was scared!

His hands instinctively reached for the sides of the hammock and he pulled himself up. He sat over one side of the colorful hammock, which reminded him of the one his abuela had hanging in her backyard. His headache backed off as his curiosity overtook him. 

He was in a hut made of polished sticks pressed together in intricate design. Aside from the hammock there was a small table, two chairs and some large clay pots. On one wall were two small, round windows without panes. Jake could see foliage rustling in the breeze. He carefully stood up, letting the hammock swing him to his feet. He walked over to the window. He noticed that the pots had water in them, with two small cups hanging off hooks. He dipped one cup into the pot and drank. The water was deliciously cold and pure. He looked out the window, amazed to find that he was actually high up a tree. This was not a hut, but rather, a treehouse!

Jake looked down and estimated that he was at least 20 feet off the ground. Now Jake did feel like he was on some movie set. He remembered being a little kid and going to Disneyland with his father. They came to the huge artificial treehouse from the Tarzan movie and they both climbed up it with loud enthusiasm. Up rope ladders, across bridges and back down to the other side where there were musical instruments to bang and scientific objects to be examined. 

Jake hated it when memories of his father overtook him. So he concentrated on his surroundings and noticed, in fact, that he was not in some fantastical village in the trees, but more likely, in prison. And with that, Jake determined he needed to escape. But how?

He heard some muffled voices. He spun around to see that there was a door on the other side of the hammock. The windows were too small for him to squeeze through, and even if he could, he doubted he could make it safely to the ground. There was no where to hide, which might have been a good idea if it weren’t for the fact that he was obviously placed in this hut on purpose and his absence would be noticed.

The door opened and Jake cringed a little, for no good reason other than fear. Jake had loved his father dearly, and sorely missed him, but they both had realized that Jake had definitely NOT inherited his father’s bravery in the face of danger. Better to avoid danger rather than face it. 

In through the door walked two kids. One was a teenage girl dressed in a simple cloth tunic. Next to her stood a boy younger than her, also in a beige tunic. They were both smiling.

“Good, you are up. We were worried about you,” said the girl as she stepped towards Jake to examine his head. She spoke with an accent, but otherwise, it was perfect English. Up to now Jesse had been assuming that they were on some small island off of Puerto Rico, where his aunt, the Captain, had been taking them. Perhaps the hurricane threw them further off course than they had realized.

Jake found the girl’s smile welcoming, not threatening like he had expected. He let her examine the bandage around his forehead. She was taller than he was and bent over a little to examine the bump.

“This looks fine. You are lucky, you know. If we hadn’t seen you in time, they would have killed you”

“Who are you? And where am I? And who would have killed me?”

“Lot’s of questions, eh? First, you will be fine here. No danger. I am Teena and this is my friend, Oto. We are the leaders of our tribe.”

“Leaders! But aren’t you too young? I mean, really, how old are you?”

“More questions?” laughed Teena. “Ok, but to figure out our ages, you’ll have to solve this riddle:  Oto, Oto, my trusted friend, is now half the age I was back then. But at that time, as we both know, he was a third of the age that I am now.”

“That’s impossible to answer,” grumbled Jake, irritated at the game.

“No it isn’t if you only think of integers.”

Chapter 7: Chasing

After Jeanie cracked the code to the stone door, Jesse, Miguel and the Captain pushed up against it and it swiveled open with remarkable ease. Cynthia stood to the left of the group and hugged Justin close to her. When the Captain motioned for the group to follow her, Cynthia shook her head. “We don’t know what we are doing, Captain. I’m not going to risk my son’s life for this crazy chase.”

“Suit yourself, Cynthia. I’m not giving up on my nephew. Anyone else want to stay back? Go ahead.” She entered into the cave with Jesse by her side. Miguel looked at Marge and Jeanie, then they entered as well. Justin looked at his mother, perplexed. 

“Mom, we can’t just stay here and do nothing. Besides, there is probably more danger here than inside the cave. Let’s go.” Just as he finished, though, they both heard an ominous grinding sound as the stone door started to close. Cynthia looked at Justin, who was tugging at her sleeve. “Come on, let’s go with the group.” She relinquished and they both squeezed into the cave just before the entrance completely shut up.

Cynthia expected complete darkness, but was surprised by the amount of ambient light. The walls glowed a golden tone. She touched the stone. It was cool to the touch. Justin was walking ahead of her, calling for the Captain. She followed.

The group of seven walked in amazed silence. Justin traced his finger along the glowing walls. This was not a cave as much as a tunnel. It seemed handmade. Where could it be leading?

The Captain stopped and cocked her head to one side. “I think I hear foot steps. Maybe it’s Jake.” She started running. Jesse ran alongside her, while the rest of the group jogged behind. Jeanie and Marge ran with ease, alongside Miguel. Justin could have kept up with the Captain, but stayed back with his mother, who hadn’t run in many years and was huffing and puffing from the exertion.

The tunnel started to slope downwards. There was a slight breeze, reminding Justin of the breezes he felt when the subway trains would arrive to their stations in New York City. He realized that probably meant there was an exit up ahead. As he turned a corner, he saw the glimmer of light ahead. He saw the Captain and Jesse standing in the exit. Then Miguel, Jeanie and Marge caught up and stopped with them. He ran up a little ahead of his mother to see why they had stopped.

The tunnel opened up into a crater-like valley, surrounded by steep cliffs and full of lush, verdant vegetation, lakes and streams. There was a red stone road leading towards the center of this valley, where it entered into a small jungle about a kilometer and a half from where they stood. The air was misty and quite a bit warmer than it had been on the other side.

Cynthia was bent over, braced against her knees, catching her breath. Jeanie was alongside her doing the same. The Captain turned to the group and said “I’m pretty sure I saw someone go into that jungle from this path. They were pulling a cart of some sort. I’m going to run down there with Jesse. Miguel, you stay back and take care of them.” She pointed to Justin and the others.

“No way, we’re all going together.” said Miguel, looking at Marge. She nodded in agreement. Justin looked at his mother, not sure she could run fast enough. 

“Well, ok, we’re off, then.”

The Captain, Jesse and Miguel started running at a brisk pace. Jeanie and Marge were right behind them. Then came Justin and his mother. It became evident, though, that they were all running at different paces. After about 10 minutes of running, Justin saw the Captain enter the jungle with Jesse and Miguel. Jeanie and Marge were about 300 meters behind her. He figured he and mother were another 120 meters behind them. He was not tired in the least and his mother seemed to be holding up ok, all things considered. 

Your Task: Assuming that the three groups of people were running at a steady pace the entire distance from the tunnel to the jungle, how far behind will Justin and his mother be when Marge and Jeanie enter the jungle? (hint: it won’t be 120 meters) What were the jogging speeds of each of the three groups?

Chapter 6: Where are the traffic jams when you need them?

Jake tired of his aunt’s tree obsession, so he edged away from his group and followed the gravel road into a forest of humongous trees. He was curious to find out what Jesse and the others had discovered.

The forest was dense, overgrown and quite dark, but not cold. The gravel road cut a clean, straight path through the vegetation. The same butterflies they had seen on the way from the beach flew around. They flicked their wings effortlessly.

Jake heard rustling branches up ahead. He stopped in his tracks. Was it really a good idea for him to be alone in this unknown forest? While his father was alive, they had spent their summer vacations backpacking in forests all over the country. He felt confident in his abilities to take care of himself in the wilderness, but as his father’s mountaineering accident proved, not everything can be predicted or controlled. Jake suddenly regretted heading out so far into the forest. He called out to make sure that is wasn’t Jesse and his crew coming towards them. No answer. But there was an increased rustling in the bushes and it was coming closer. Jake turned to run back to his Aunt’s group. He saw the clearing up ahead just as some sort of fabric came down over his head and engulfing his entire body. He tumbled to the ground, scraping his knee and hitting his head on some rock. The blow knocked him unconscious. 

Back at the tree, Justin was the first to notice Jake’s absence. The Captain frowned a little, then she said “Oh, he must just have gone a little ahead. Why don’t we try to catch up to him?”

About 20 yards from the forest’s edge they heard some scuffling noises. The Captain took off in a sprint, with Justin and his mother just behind her. She yelled out Jake’s name. No answer. She disappeared into the forest. When they caught up, she hunched over the gravel road examining marks in the ground. Justin thought she might be crying, but when she looked up at them, her face was a mixture of anger and fear, but no tears.

“Something’s happened to Jake. I think he was attacked, but there’s no sign of major violence. No blood or anything. I would dare say he’s been dragged away in a sack or something. At least, that’s what this mark seems to be.” She pointed to a long, shallow trough leading away on the gravel road. “But there are so many strange marks here. Some three toed foot prints, for example. I don’t get it. I didn’t say anything before, but this whole place is creeping me out. Strange marks on the cliffs, funny grooves in the road and now this. But we need to move quickly. This just happened, let’s hurry up, they might be just ahead.” The Captain got up and started jogging down the gravel road.

Justin looked at his mother. They were both scared now, but started jogging behind the Captain. Justin looked into the vegetation. What if it were some animal doing this? What if Jake had been attacked by some fierce carnivore, say a tiger (are there tigers in the Caribbean?) or some other, yet unknown animal. He knew that the Captain was a woman of action, but he wanted to slow down, get out of this forest and think out their options. He knew his mother would agree with him, but they both felt they shouldn’t just abandon the Captain.

They continued down the road for about 10 minutes when they heard Jesse and Miguel yelling up ahead. “Maybe they have Justin with them?” he thought, but as he turned the corner, he saw the Captain stop in front them.  They shut up immediately.

“Look it, guys, something’s happened to Jake. He’s been taken away. Stop bickering and focus on what’s important, will you?” They two men nodded. Then Miguel said,

“Captain, we were just looking for you. We found a massive stone door with a secret code. We just saw it close, but we didn’t see who went in. Maybe they have Jake.”

Jesse led them to the door. It seemed to be protecting some cave in the side of a mountain. Marge was studying some markings etched into the stone.  “It’s some sort of code, like a key. I think that the first four drawings present some sort of pattern that has to continue to the next three. I bet if we get them right, the door opens. My only hope, “she added, “is that if we get them wrong, nothing bad happens” She looked at the Captain ominously.

Your task:  Break the code! If you continue the pattern, what will the next three figures look like? There might be several correct answers to this puzzle, so try to find them. Explain your thinking carefully.

Chapter 5: Reuniting

The Captain led her group up the beach towards the pass. It was strange that such a beautiful beach would be untouched and pristine. She had a hunch, though, that just ahead they would find civilization. Cynthia, Justin’s mother, walked alongside her, while the boys hung back. Cynthia wouldn’t stop talking.

“What if we don’t find anything? What’ll we do out here, all alone? What if we get hurt? I mean, really, do you have any idea where we are?”. 

“In fact, I do know where we are.” The Captain was lying. “Just before the storm I had charted a course towards Puerto Rico. I had Jesse check the instruments just before we crashed. We weren’t far away, that I know. In fact, this may well be Puerto Rico right here”  

They arrived at the pass and found, indeed, that it was a man-made road leading down the mountainside to the beach. Cynthia relaxed, but the Captain tensed a little, without showing it. Something was very odd about this road, but she couldn’t quite figure out what it was.

The road was unpaved and covered with a reddish green gravel. What struck the Captain odd was the fact that there were three grooves worn into the road instead of the expected two. There were odd scratch marks on the cliff wall, as well. They looked like writing, but the Captain was unfamiliar with this alphabet. The failed to notice.

The road was remarkably steep, but passable. Cynthia wondered out loud what type of vehicle would be able to use it. The group walked a bit, then rested, then started up again. After about an hour, they came to a turn in the road where it suddenly leveled out. They were just about to continue along the road, into the forest, when Justin cried out: “The flare...I see the flare!” Over the tree canopy was the unmistakeable flare of the other group. Judging by its trajectory, they were not too far apart. The road appeared to go in the right direction, so they chose to stay on it. Maybe there was a town up ahead. Jake and Justin started walking ahead of the women. Cynthia looked relieved. 

The gravel road gave way to a dirt one, always with the same three grooves in it. Forest around the road was thick with tropical like vegetation, although the temperature was really quite moderate. There were occasional hoots and hisses from deeper within the forest. Several extraordinarily large and colorful butterflies hovered lazily overhead. The boys were sprinting ahead with the Captain walked determinedly alongside a cheerful Cynthia.

They came to a circular clearing, at least a kilometer in diameter. In the center was the largest tree that Cynthia had every seen. It seemed to endlessly reach towards the sky as a long, thin pillar of wood and bark. She turned to the Captain and asked if she knew what type of tree it was. “Some sort of pine, but I’ve never seen anything like it.” 

As the approached the tree, with Justin racing Jake around the trunk, the Captain looked at the shadow extending across the grass (which oddly enough, seemed cut like a lawn, not natural). She paced along the shadow and counted almost 240 strides from the base of the shadow to its tip. She went over to Cynthia, stood straight up, sort of parallel to the tree, and asked her to count the strides of her shadow. Cynthia counted 3 strides.

“Now I know about how tall this tree is, and let me tell you, it breaks all known records.” 

“How did you figure that out?” asked Cynthia.

“Well, I’m 5 feet 6 inches tall and when you told me how many strides long my shadow was, I did a relatively easy calculation in my head. It’s all estimation, of course, but based on solid math principles. Now I want to figure out about how old this tree is.” She called Justin and Jake over to them. “There’s this old trick I learned in school to estimate how old a tree is by finding its circumference and thinking of its growth rings.  Cynthia and I are about the same height, Jake, you seem to be about 6 feet tall, right?” He nodded. “Justin, I’m guessing your a little over 5 feet. So here’s what we’re going to do. Each of us is going to hug this tree and touch the finger tips of of our neighbors.” Amazingly, the four them just managed to touch their fingertips, meaning they formed a perfect ring around the trunk. The Captain sat pensive for a moment, then scratched some numbers in the dirt. After a little while, she stood up and announced that she knew that this tree was likely over 2000 years old.

“2000 years olds! That’s impossible. A tree can’t live that long, Aunt Patty.”

“Actually, you’re right, Jake,” interrupted Justin. “It’s really only about 1000 years old.”