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.”

Chapter 4: Salvaging

They had survived!

Justin was soggy and cold, but at least he was alive and unhurt. So were his mother and the other passengers from the sailboat. The previous night’s storm had thrown their sail boat against a rocky outcropping just off shore. They managed to scramble ashore as the winds and rain abated.

The captain and her teenage nephew were scouting the beach for useful debris from their shipwreck. Random bags and boxes had floated onshore near them. The boat lay in ruins on the rocks. It finally struck Justin that they were shipwrecked for real. Three weeks earlier Justin had begun planning for the start of his 8^th grade year. Then, out of the blue, his mother told him that they were going on a chartered Caribbean sailing trip with people he didn’t know. While on this trip, a storm overtook the ship. And now here he was, stranded with his shipmates. It felt like some bad t.v. movie plot.

Justin’s mother held onto him silently. Jesse, the weather guy, stood up to help the captain. The three college students, a guy and two women, sat huddled against a stone. Justin could hear the captain speaking with her nephew and Jesse, but the rest of them just stayed silent.

The Captain, her nephew and Jesse came towards them with several sacks salvaged from the ship. 

“Look” she said, “we have to accept the fact that we’re stuck here on this island until we can figure a way get off. Jake, Jesse and I were talking and we think we should explore this place. An island this big has to be inhabited by people somewhere. There must be some sort of town or something over in that direction.”  She pointed up over the red clay cliffs to the green mountains. “We’ll split up into two groups: Jake, Justin and Cynthia will go with me with up the beach and out over that pass in the distance. Jesse, Jeanie, Marge and Miguel: you guys will head up over there, “she pointed to what looked like a crack in the cliffs that formed a sort of pathway up into the mountains. Justin saw Miguel get all serious, even a little bothered, by what the captain said, but kept quiet.

The Captain and Jesse both emptied out the sacks they had carried over to the group. Out fell rope, food packages, canteens and random other items they had deemed useful. Jesse took up one of the empty sacks and said, “These sacks can be fashioned into a sort of backpack. We’ll split up these items between the two groups. We figure that when one of the groups finds civilization, it can send out one of these flares.” He held one up to show them. 

“This is getting more like a silly t.v. show every minute,” Miguel muttered. Jesse ignored the comment and continued. Everyone else started to divvy up the stuff between the two sacks. Jesse approached Justin with two large canteens and a cup. 

“Hey Justin, I have this task for you. In this first canteen I emptied a liter bottle of water. In this other is a liter of Gatorade. Both of the bottles were starting to break.  We want to split the liquids up evenly between the two groups but we only have these two canteens to carry them. We’ll have to mix every thing up.”

“How will we know he’ll do it fairly,” interrupted Miguel with some severity. Justin grew a little nervous at Miguel’s apparent mistrust of him.

“That’s easy,” replied Jesse. “All you have to do, Justin, is pour out a cup of this Gatorade, mix it into the water canteen, then pour out a cup of that mix and put it back into the original Gatorade canteen. Both canteens will have not only the same amount of liquid, but also the same concentration.”

Your task: Is Jesse correct? If this is true, does this method work for any volume. If false, is there a better way to do this with the materials available to them?

Chapter 3: Sailing

Five days into the sailing trip Jesse was having serious regrets. What had begun as unbearably exciting had become torture. Barely an hour after setting out to sea, his normally baritone voice went quiet and he sat pale faced in one corner of the boat. He felt fuzzy headed and nauseous.  After two hours, he started vomiting. The captain and her assistant initially laughed off the incident as one more passenger working to earn his sea legs. To his chagrin, nobody else showed any sign of sea-sickness. Even after his stomach calmed down, his head continued to pound. Jesse decided that laying down in his bunk was his only option. He basically stayed there for entire days at a time, only venturing out when they made landfall at some island.  

It wasn’t supposed to be this way. He had just accepted his first real job as a meteorologist for the local television station. It occurred to him to book this sailing cruise to celebrate his new career move as well as to acquire some important real world weather experience that he was lacking after living in academia for so long. The wrenching lurches and lunges of this boat quickly took away both the celebration and the real weather experience.

Jesse knew that September was a very interesting time in the Caribbean. The weather was balmy but fresh breezes hinted at changes to come. The ocean was water was its warmest temperature of the year.

The Captain stopped laughing off Jesse’s illness. She was serious when she brought Jesse water and some Dramamine. She insisted he come above board and take the helm of the ship as they sailed.  She told him to set his sight on the horizon as a way to combat the nausea and headaches. She set off to work on the sails. 

He looked out over the horizon. It was no longer the light blue contrasting the deeper hue of open sea. Dark clouds were gathering quickly. The breeze turned into a wind and the sea grew choppy. The Captain and her nephew started tying down equipment, tightening the sail and telling the other passengers to go inside.  Jesse knew that some serious weather was coming their way and he cursed himself for not looking into it more seriously before booking this trip. If anybody on this ship could predict the weather, it should be him.

The captain came hurriedly towards Jesse, wet from the waves crashing in on them from all directions. 

“We’re in trouble. We need to send out an SOS to any ships nearby. Go down in the cabin to the radio. You should also check the radar to see if there are any land masses close by. Move it!”

Jesse scrambled down the hatch. Sea sickness had been replaced with dread.

Your task:

Radars are radio wave sent from a radar station and reflected by the target to the station, causing a spike to appear on the wavy line. The indicator is marked so that the number under the spike shows the distance in miles from the station to the target. The left screen supplies data from the shore radar station at A, the right screen from B. How can you use the 75 and 90 numbers on the indicators to locate a target? Exactly where is the target on an extension of this map? Draw this map. What assumptions did you need to make to solve this problem.

Chapter 2: Loading

Patricia sat on the edge of the sailboat watching a group of people walking slowly down the dock with their baggage in tow. This was going to be the last sailing trip she would captain for a long time. After this two-week trip around the Caribbean, she would be handing over the keys to the boat’s new owners and setting off to her new life. If only she knew what that was.

Even after 10 years guiding one group of vacationers after another between the islands, she had not lost the sense of excitement before each trip. What awaited them just over the horizon was both known and mysterious. She was sure she had visited every island and rocky outpost between the Puerto Rico and the Mexican coast.  But deep down she knew that the ocean was far deeper and secretive than it let on to the casual observer.

This summer had treated her well. Her trips had been booked earlier than normal and her clients had been a cut above the rest in terms of their competency and sense of adventure. In addition, she had developed a very close relationship with her 17 year-old nephew. When her sister had called her last spring and suggested that she offer Jake a summer job on her boat, Patricia initially balked. She hardly knew her nephew and he had never spent time on a boat. But her sister needed this favor and so she relinquished. To her surprise, Jake was a quick student and enthusiastically took on the responsibilities of ensign. Soon she wondered how she had ever sailed without him.

“Aunt Patty, I finished putting away the supplies.” Jake sat down next to her to watch the six passengers struggle down towards them. “Should I go help them?” he asked. Patricia nodded negatively. She had long ago learned that for these trips to be successful, the passengers had to understand they would be responsible for their own stuff. But why did they always bring so much?

They approached the boat. Patricia and Jake stood up and waved to them with huge smiles on their faces. Patricia recognized the mother and her son as they came forth first. She was very impressed with the tiny suitcases each had in their hands. 

“Hi, you must be Cynthia and Justin. Welcome to the S.S. Gobio.” She shook their hands just as the other four passengers approached. 

A tall man in his late twenties spoke out in a boomingly cheerful voice “I’m Jesse, you must be the captain?” Patricia nodded. He looked around the boat.

“Pretty cool. Even looks better than it did on your website”. 

Behind Jesse were two women and a man in their early twenties. They introduced themselves as Jeanie, Michelle and Miguel. They had signed up for this trip as one last summer adventure before returning back to their senior year at college. 

Just before Patricia was going to launch into her explanation of rules and regulations of her boat, Jesse raised his hand and said:

“You know what, guys, look over there.” He pointed to the numbers on the ship: 45254. “ You know what that kind of number that is? Palindromic. It reads the same backward as forward. The weird thing is that I noticed the same kind of number on our shuttle bus. When we got on, the odometer showed 15,951. At the time I thought it was strange and was sure that it would be a long time before that happened again. But almost exactly two hours later, when we arrived here, the odometer showed a new palindromic number.” 

Jesse looked at Justin and said, “What could be cooler than that?” Justin could think a several cooler things, but decided to keep quiet.

Your task: Based on the information provided, how fast was the shuttle traveling in those 2 hours?

Chapter 1: Packing

“Awesome!”


That was Justin’s first reaction when his mother told him back in July that he was going to miss the first two weeks of school to go with her on a Caribbean sailing trip she had booked a month earlier. But now he wasn’t quite so psyched. He felt left out from his soccer team and his friends as they chatted about their up-coming 8th grade year. He wondered what he would do without his cell phone or his laptop. He also wondered what was on his mother’s mind? It was so unlike her to disregard school and send him on such an adventure.


He wanted to go, to be sure. But why now and why with her? Did he have any choice about this trip? He sensed he couldn’t broach the subject with his mother judging by the distracted way she had sent him to his room to pack for the trip. He was both excited and confused at the same time.


In his room Justin reached under his bed and pulled the small suitcase his father had bought. He heard his mother’s cell phone ring and her muffled voice from down the hallway as she must have slipped into the kitchen. He heard her laugh. Who had called?


He was rummaging around his room when his mother called from down the hall.
“Justin, remember to pack enough clothes for the whole two weeks… I won’t be washing anything, ok?”


For many other kids his age this might have seemed like an easy task. But Justin was very particular about his clothing choices. He never wore the same clothes two days in a row, and he always washed his clothes after wearing them for a day. He also never wore a shirt that was the same color as his pants.


This presented him with a serious dilemma, one his mother wasn’t about to deal with in her current state. What to pack? To make matters worse, he was supposed to fit all his clothes, toiletries and a reading book into this tiny suitcase that only measured 24” by 24” by 12”. His father was gone on one of his business trips, his mother was mysteriously preoccupied and he here he sat on his bed frozen by his packing conundrum.

Your task: Create a practical packing plan for Justin. Determine what the reasonable number shirts and pants (or shorts) he needs to guarantee he can follow his clothing routine even on the sailing trip. You will also have to determine if all the clothes, toiletries and book will actually fit into the suitcase he has to use. Assume that his socks and underwear will use up about ½ cubic feet of his suitcase.  Create a detailed and organized list of the clothing items as well as a clear diagram (with dimensions) of how it will all fit in this suitcase.