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Saturday, October 24, 2009

Verdania: A Mathematical Odyssey: Chapter 6: Meeting

Chapter 6: Meeting
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.