Arrows at Dawn (Iasc)

[Guido Faragas]

“Hmm,” said Guido uncertainly. “That’s a bit of a conundrum. By your logic, is appears as if the runner can never catch up to the turtle, for he will only ever cover part of the distance between them. But there must be a trick here, for common sense suggests that a faster runner will eventually catch a slower runner.”

Guido thought for a few chimes and then grabbed a nearby stick. He snapped a piece off the end and measured it to be about a tenth of the size of the original piece. Then, he grabbed two stones a dark one and a light one.

“Suppose the dark pebble is the runner,” he began, laying the stone on the ground. “And this stick is 100 metres.” He lay the stick next to the pebble and then put the lighter stone on the ground at the other end of the stick from the darker one.

“If the runner runs one hundred metres he will end up where the turtle started off. Meanwhile the turtle will have run 10 metres. He moved the dark pebble to where the lighter one was, measured 10 metres with the smaller stick and moved the lighter stone forwards. “So, after the runner runs 100 metres, he is only 10 metres behind.”

“If we repeat this....” He moved the dark pebble forwards by the length of the larger stick and the lighter pebble the length of the smaller stick. “The runner is now well ahead. So, if the total distance travelled by the runner is 200 metres, the turtle would start off 100m ahead but then only cover 20 metres in the time that the runner covered the 200m. So, the turtle would be well beaten in such a race covering 200m.”

He scratched his head. “It must be the way that you posed the problem that creates the conundrum.” He laughed. “But I still can’t work out when he will catch up apart from the fact that it is after 110 metres and before 200 metres.” Guido cupped his head in his hands, thinking deeply.

“Aaah!” He returned the pebbles to where they had been after the first part of the race with the turtle 10 metres ahead. Then, he snapped the two sticks in half. He moved the dark pebble forwards the length of the larger stick and the light one the length of the smaller stick. The dark pebble was well ahead. “So when the runner has covered 150 metres the turtle will only have covered 15 metres plus its start of 100 metres. So, the point where the runner catches up must be before 150 metres, which narrows it down a little more at least. If I could continue snapping these twigs in half, I might eventually find the solution but the twigs would be very small!”

He glanced towards Iasc. “This is a very intriguing problem. I have a feeling you are teaching me some mathematics here. Is my logic in the right direction? Can the problem be solved?”

[Iasc]

“There is no trick as you suggest. It is not that type of question. The way that you are thinking about this is the way that most people would. It is very obvious that in a straight race the runner would beat the turtle hands down because he is plainly faster. For that reason there is nothing wrong with what you say, however, I am asking you to think about the case where, whenever the runner catches up to the turtle, the turtle has crawled to its new position. You must look at the race as a series of continuous segmants.”

Iasc sighed, feeling that his explanation was sub-par.

“It is a problem with only one purpose. It exists in order to help you think logically.”

He picked up the stick that Guido was using. “Here is what I have made of it so far. Let's restrict ourselves to where there is only one metre between runner and turtle.”
In the earth Iasc wrote down: 1 1/10 1/100...

“These are the next three distances that they travel”, Iasc explained, never taking his eyes off the numbers on the ground. He then causally put an addition sign between each one.

“Whether it can be solved or not, I can't say. There seems to be no final member in this sequence. If there was a final member, then I'm sure that that would be the answer. If I could find what they add up to I would have my answer.”
Iasc sighed and shook his head, still unable to solve the problem.

[Guido Faragas]

Guido watched carefully as Iasc drew the numbers in the earth. “So every number gets smaller than the last,” the reimancer said slowly, as the cogs within his mind turned the problem over and over. “Thus, you must eventually reach a number which is so small that it is indistinguishable from zero. Hmm.”

“Well one-tenth is equal to ten over a hundred, so the first two small numbers you add must be equal to 11 over a hundred. The next number would be 1/1000 and 11 over a hundred is 110 over a thousand, so adding that to the number must give you 111/1000. There is a pattern! However many zeros you get, there is one less one on the top. This will just continue on and on. Is this any help, or am I talking nonsense?” Guido’s brain was spinning. He had had to learn much logical thinking from his master to advance in the magical arts but mathematics clearly required a very different perspective. Still, he was increasingly intrigued by Iasc’s use of numbers.

[Iasc]

Iasc looked at Guido strangely. “You are very smart”, he noted
He looked back to the ground, extremely surprised by his companions mathematical abilities.
“You have hit my theory bang on the head...sort of. If you continue to add, 111/1000 becomes 1111/10000 which becomes 11111/100000 and so on. Each consecutive number will get incrementally larger as you continue to add numbers, but the difference between two consecutive pairs of numbers will get smaller and smaller until it gets close enough to zero to be negligible. I think you said that the numbers get smaller instead of larger.” Iasc gave a shrug.

“Anyway.” He picked up the stick again and wrote: 1111/10000 – 111/1000 = 1/10000
11111/100000 – 1111/10000 = 1/100000

“As you can see, of these two answers, the former is the larger.” He demonstrated this: 1/10000 – 1/00000 = 9/100000

“So, since each number is getting smaller it will soon become close enough to zero.” He was talking faster and faster now, his hands gesticulating because of the excitement he gets from explaining theories to people.

He calmed down when he had finished and leaned back to look at the work. “Honestly, if you can do this I'm not sure there's much I can teach you with regard to your business deals. Although I do like a challenge”, He smiled.

[Guido Faragas]

Guido gave a modest shrug. “I think that it was beginner’s luck rather than any great ability on my part. And, you explained the problem well, so the solution was almost inevitable! You must teach me more in the future, for I find it fascinating. Perhaps, I am an undiscovered mathematician.” He laughed loudly.

“Well, I guess that I should be returning to the city – I have work to attend to. Do you wish to take up my offer of a trip to the graveyard?”

OOCI guess we are nearly finished up here. If you’re up for the graveyard dig, then I’ll open a new thread soon

[Iasc]

"We may soon see", Iasc laughed along with Guido.
"I will go to the graveyard with you", he said with a hint of excitement in his voice which he couldn't quite disguise.

As Iasc watched Guido walk off towards the city he began to look around him at the trees, whose leaves were beginning to turn a beautiful golden brown. He lay down on the ground and shielded his eyes from the sun, letting the cool air wash over his body, basking in the wonders of Caiyha's domain.

OOCCould you wait a few days before starting the graveyard thread since I've been busy lately and have a bit to catch up on. PM me when you've got it up

[Guido Faragas]

Guido made his way back to the city in his usual slow, lethargic manner, pondering on the day’s events. It had been an interesting day – archery, mathematics, stories and philosophy - a potent combination. He hitched his bow further up on his shoulder, nodding a quick hello to the familiar guards who stood at the gate.

OOCThat’s fine – I’m likely to be busy the next week or so as well , so I’ll leave it a couple of weeks before opening a new thread. Thanks for the fun thread!

[Leviathan]

Thread Award! Iasc   Skills Skill Gain Info. Weapon (Shortbow) 3 Aiming and firing with the weapon Mathematics 2 Visualizing angles, Solving problems Gambling 1 Casual betting Storytelling 2 Telling about personal history, Philosophical tale Philosophy 2 Philosophical tale and thought Teaching 2 Sharing knowledge of mathematics   Lore The Open Stance (Shortbow) Short Stories: Guido and his Bow Philosophical Stories: The Turtle Race Guido   Skills Skill Gain Info. Observation 1 Observing the lack of wind Weapon (Shortbow) 3 Aiming and firing with the weapon Gambling 1 Casual betting Storytelling 5 Telling of tales Mathematics 1 Solving Problems   Lore The Open Stance (Shortbow) Short Stories: Guido and his Bow Philosophical Stories: The Turtle Race

Additional CommentsVery interesting thread guys. I'm sorry I deleted this from my office initially, which is why I didn't get to it in a reasonable time. I'm not sure how I made the mix up. If you have any questions, comments, or concerns feel free to PM me and we can find a solution.