Teacher went through the tablets one by one, then called out the names of the pupils who had the right answer, only about five out of thirty or so students in the class. He turned over the final tablet, Carl’s, his expression changing to one of amazement.
The answer was right. But how had this young boy done it so quickly? He begrudgingly called out Carl Friedrich Gauss’s name as having the correct answer.
“So how did he get the answer just like that?” Tollie asked Mouse.
He scrunched up his face, deep in thought.
Mouse tried to help him out. “Think of a mathematical way to get the answer instead of just adding up the numbers one-by-one. Think of the list of numbers itself. Can you see anything about it that could lead to such a quick answer? There’s more than one way to skin a cat!” Again, Mouse seemed to relish the last – he and cats would never be an item, it was clear.
Mouse was generous with his hints, but Tollie could not get over his awe of young Carl Friedrich’s amazing feat. He just did not see how it was possible.
To his astonishment, Carl himself walked over to them, seeming to see them while no one else had (for he truly had uncanny ability), and introduced himself. Mouse and Tollie greeted him warmly. Of course, the curious Tollie could hardly wait. “How did you do it, Carl?”
“It was a matter of pairing up the numbers,” Carl explained. “You have one-hundred numbers in sequence from one to one-hundred and you have to find the sum of all of these. I saw at once that you could pair up the numbers into fifty pairs in a certain way.”
Tollie was beginning to get an idea, but it still did not fully dawn upon him. Mouse, of course, already knew the answer, which Tollie found irritating at times.
“You can make a pair out of the highest and the lowest numbers,” Carl went on. “That is, 1 and 100, which add up to 101. The same is true of the second highest and lowest numbers, so that 2 and 99 also add up to 101, and the next highest and lowest, 3 and 98, also add up to 101, right down to 50 and 51, which add up also to 101. This is a process called extrapolation, the extending out of a pattern that proceeds in a logical way. If one step is true, the next step will also be true, and the one after that, and so on.
“Soooo, what you have are 50 pairs of numbers that add up to 101. The total would be 50 times 101, which gives you 5050.”
Tollie now understood and was chagrined that he had not thought of this himself. “You make it seem so easy, Carl.”
Carl smiled at Tollie. “I always believe there are answers, it is just a matter of finding them. But believing in the reality of there being an answer is always the first step. Even if there is not an answer, you have to show why not. And you can think your way through it, by patiently applying logic and reason to the question at hand.”
“I see, somewhat,” Tollie said. “I didn’t really believe there was an answer to this one so that worked against me finding one. And, then, I did not think to prove that an answer was not possible, which would itself be an answer of sorts.”
“Exactly,” Carl said, his eyes twinkling. He took his leave of them. “I’m supposed to be sitting at my desk, but it was fun meeting you, Tollie. I think your natural curiosity will make you a better mathematician than you think right now.”
“And it was fun meeting you, too. You taught me to believe in and really seek answers even when it seems at first to be impossible. Thank you!”
“You’re very welcome.” Carl gave a slight bow, which both Tollie and Mouse returned, and wandered back to his desk.
“That was wonderful, Mr Mouse. There is so much to learn here!”
“Oh, yes, Tollie, it is a big world. Not quite like your everyday world, is it?”
“But somehow similar.”
“Oh, yes, that it is,” Mouse said wistfully, for it was the only world he existed in, after all. “Just remember, not everything you find here is the exact truth. For example, this incident with Carl and his teacher probably never really happened, but when enough people believe it happened, it can be real here in the World-Wide Web. We don’t really know, but it’s doubtful and many think it is a made-up story. But it is a good story and that is why it is real here. Just take things with a grain of salt.
“Nevertheless,” Mouse added, “what Carl told you is quite true and real. You do need to believe that there are answers and that you can find them. And when there is not an answer you can show why not. And, remember, when I give you a challenge it will always be an honest one, and there will be an answer that you will be able to figure out.”
“But I haven’t figured out any of them so far,” Tollie protested.
“You are still learning, and you are getting better at it. Did you notice how quickly you caught on when Carl explained his method to you?”
“Sort of,” Tollie said, still doubtful.
“You will see,” Mouse continued confidently. “Shall we go? Just let me know where…”
Tollie put his hand on Mouse’s back and thought he would like to see the Great Pyramid in Egypt!
CHAPTER 5
IN THE SHADOW OF THE GREAT PYRAMID
And they were there.
“This takes some getting used to, Mr Mouse!”
They were at the foot of the Great Pyramid and beholding it with a certain healthy respect. Seeing the real thing gave Tollie the true measure of it. It was, simply, ‘huge.’
It inspired Mouse to sing:
“Happy, happy are the mice.
Happy are the nice, nice,
nice, nice mice…”
He danced around joyously. There was in front of them also, of course, the Sphinx, a cat of sorts, practically in the shadow of the pyramid, and Mouse seemed to flaunt his “mouseness” (or “mousity”), aware that the Sphinx was only a statue and could do him no harm. Tollie chuckled to himself.
They heard another voice singing, joining in with the same tune as Mouse’s signature ditty.
“I am, amid a pyramid,
amid a pyramid, mid
pyramid…”
Materializing before them was an Ancient Egyptian. This seemed sudden to them, but it was only because this was the moment the Ancient Egyptian had chosen to reveal his profile to them! You see, he was completely flat and invisible when he faced you. Only when he turned his side to you in profile could you see him.
Tollie had thought it was just that the ancient Egyptians had not mastered perspective and could not represent three dimensions on a flat surface in their art, so they appeared two-dimensional. But here, they actually were flat and only visible in profile.
“They really were flat?” Tollie asked incredulously, and Mouse grinned broadly.
“That is the only way we can see them here.”
Tollie was getting an inkling of the nature of this odd virtual world.
Ancient Egyptian looked at them hesitantly, deciding something in his mind. Evidently, Tollie and Mouse were friendly enough in manner that he went ahead and asked them for help with a problem he was having.
“Pharaoh sent me to find the height of the Great Pyramid and I cannot climb it to measure any of its sides. It is too smooth. Also, I cannot measure its height directly because its top is over the center. How can I find out how tall it is?”
Tollie looked up at the steep inclines of the Great Pyramid, noting how high it was and wondered how to find out just how high. “Well, Mr Mouse, do you know if it can be done?”
“Oh, yes, there is a way,” Mouse assured him. “Remember what Carl said. If you believe there is an answer it will be possible for you to find it.”
“But I can’t climb that pyramid either.”
“Maybe you don’t have to climb it.”
Mouse seemed to know everything. Tollie again found this irritating, but he had to give Mouse some leeway. He was a widely-traveled mouse and it was only his job.
“An indirect measure, you mean?”
“You’re on the right track,” Mouse encouraged him.
Tollie thought out lo
ud. “What if there is a way to measure it indirectly?”
He thought and thought. He was becoming more relaxed about thinking through these new challenges. His face was not nearly as scrunched up as the last few times.
“You could measure something else that would give you an idea, something that you could measure directly? That would give you an idea of how high the Great Pyramid itself is.”
“But what?” asked Mouse.
“Hmm…”
The sun was beginning to set and the air was cooling as the shadows grew longer. Tollie glanced at it, squinting. A light dawned on his face, the light of understanding, not the sun’s rays, though they had something to do with it.
Mouse was smiling happily, humming his little tune, “Happy, happy are the mice…”
Tollie looked around. He found himself holding a yardstick, as he had thought that he would need one! Strange place, this “virtual” reality!
“Now you can measure, Tollie,” Mouse hinted, though it seemed a somewhat trivial and obvious hint. He had a yardstick so he could measure. So what?
“But,” Ancient Egyptian interjected, “what can he measure? Certainly not the height of the Great Pyramid.” The poor fellow still had a worried look. Clearly, he did not want to disappoint Pharaoh and wanted to return with the required measurement.
Tollie patted his shoulder. “I have an idea that will help you.”
Ancient Egyptian turned his profile to Tollie and smiled a smile that Tollie knew was a happy smile even though he could only see half of it.
“We are going to measure the shadow of this yardstick,” he said. “When the shadow is the same length as the height of the yardstick, we will measure the shadow of the Great Pyramid, which will at that time also be the same length as the height of the pyramid!”
Ancient Egyptian considered this, then smiled, being careful to keep his profile turned to Tollie. “Yes, yes, indeed, that would do, oh, yes, indeed!”
Tollie was gratified to see the worry lift from Ancient Egyptian’s face.
Mouse danced to his own tune in the background. He was celebrating. Tollie had solved his first puzzle!
“Tomorrow, when it is early and the sun is at the right angle, I will measure it!” Ancient Egyptian announced. “I cannot thank you enough! Now I will be able to go back to Pharaoh with the measurement of the height of the Great Pyramid itself!”
Tollie smiled. He had solved the puzzle this time and it felt good. Ancient Egyptian was overjoyed. Mouse was still dancing to his little tune, but he winked at Tollie. It seemed to Tollie that Mouse was giving him due credit for solving Ancient Egyptian’s dilemma. Tollie had to admit to himself, with a certain pride, that he was getting better at this.
Ancient Egyptian thanked them and wandered off, to do his measurement the next day and return to Pharaoh, having accomplished his mission successfully.
CHAPTER 6
THE GRUMPY CAMEL AND THOUSANDS OF BANANAS
Mouse tapped Tollie’s shoulder. “See that camel trotting over? I think you will be facing a new challenge anon.”
Tollie saw Camel stopping by a large pile of bananas that he had not noticed before.
“Hello,” Camel said in a gruff voice, sounding something like a snort. His wide nostrils even flared.
“Hello, Mr Camel,” Tollie replied formally, in case, like Mr Mouse, Camel was a stickler for being properly and formally addressed. “I’m Tollie. You can just call me ‘Tollie.’” He smiled and held out his hand, then realized that, unlike Mouse, who could shake hands with his hand-like paw (even as he stood and walked on only his two hind legs), this was just a regular four-footed talking camel. He looked grumpy, but Tollie dismissed this. He thought all camels looked grumpy.
Camel looked at the horizon. The desert stretching toward it went on endlessly. The vastness of the distance seemed to discourage Camel, for he snorted again, a very resigned sounding snort. “See these bananas?”
Tollie and Mouse both duly nodded.
“A thousand miles in that direction is an oasis.” He indicated the westerly direction where the sun was now setting. “At that oasis are many hungry camels. They need all of the bananas I can get to them. But, I have to eat a banana for every mile I travel. I can only carry a thousand bananas at most. So, if I took a thousand bananas, I would get there with no bananas at all. They would all be used up to make the trip. Not a good thing. There are three thousand bananas in this pile. I can take a thousand and set out and I can drop some bananas at any point in between here and the oasis and come back to get more bananas. I can do this again and leave piles of bananas at different places in between. I need to get as many bananas as possible to the camels, they are very hungry. How can I get the most bananas possible to them?”
Camel looked grumpier than before considering this.
“Tollie here is good at solving problems,” Mouse put in at this point. Unlike his feelings toward cats, he had nothing against camels. He had nothing for them either, but at least nothing against them. “I’m sure he can help you.”
Tollie was not so sure and wished Mouse had not volunteered his services. He had no idea how, starting with 3,000 bananas, taking only a 1,000 at a time, and eating 1 banana for each mile he traveled, Camel could get the most bananas across the desert to the oasis 1,000 miles away.
Mouse was just peering at him unhelpfully.
“Mr Mouse,” Tollie implored, “you always know the answers. Don’t you have even a hint?”
Mouse grinned. “Happy to help, Tollie. Think of leaving with 1,000 bananas, dropping some off somewhere in between, and returning to pick up another 1,000 bananas and setting out again, only to once more drop off some bananas somewhere in between. Remember, on the second trip you can use the bananas you dropped off on the first trip. Then you can return and set out on your third trip, taking the last 1,000 bananas from the start point. Count up the trips, Tollie. That’s my hint to you.”
It was a lengthy hint, but Tollie was grateful for any hope of getting a handle on Camel’s problem. He felt sorry for Camel, as he seemed genuinely concerned about his fellow camels, hungry at their oasis. They had water, but no food!
Camel watched hopefully as Tollie began thinking. He knew Tollie was thinking because of the way his face was scrunched up. Tollie was getting used to thinking things through, but this was a tough one and he had reverted to scrunching up his face while thinking about it.
“One trip out, one back, one trip out again and one back again, then one out and that’s the last outward-bound trip, and the last trip of all. Five trips total.” Tollie pondered this for a while, having followed Mouse’s hint.
Camel watched him expectantly. Mouse had a little smile on his face. “Think about how many trips starting at the second thousand bananas, too,” he hinted.
Tollie had hardly begun thinking about the first trip, but now he took only a few moments to grasp the implication. “Three. You already made one trip out and one back, two trips out of five.” As he said this, he began to see a possibility of getting an answer to how to get as many bananas across as possible.
The answer was there, he just had to puzzle it through.
“Five trips from the start, three trips after picking up the second bunch of bananas, only one trip out with the third, last bunch, five trips in all,” he mused. “Maybe fifths and thirds…”
He thought on it. Mouse waited, an encouraging gleam in his little round eyes.
“All right, I have an idea.” Tollie took a deep breath.
“Start out with the first 1,000 bananas. Go one-fifth of the way, or 200 miles, because you have four trips ahead of you still. You will have to pass this same 200-mile distance five times using up 1,000 bananas in all just for this part of the trip. And, you need to leave enough bananas so Camel… er, Mr Camel… has one banana to eat for every mile he travels. Soooo… he can go out to the 200-mile point, which is 1/5 of the 1,000 mile distance and leave 600 bananas, eating 200 going out and 200 going back,
using up only 400 of the first 1,000 bananas.”
Mouse was doing a bit of a dance and humming quietly in the background, almost to himself, knowing Tollie was doing his thinking right. This encouraged Tollie to go on.
“Now, Mr Camel sets out from the start point for the second time, with the second thousand bananas, leaving behind the last thousand bananas. He has three trips ahead of him and 600 bananas waiting for him at the 200-mile point. When he gets to the 200-mile point he has 800 bananas since he had to eat 200 bananas to get there. He picks up 200 bananas, giving him his maximum load of 1,000 bananas again. That leaves 400 bananas at the 200-mile point.”
Camel seemed to hearten, seeing that there was some chance here of getting the most bananas he could to his fellows at the oasis. Tollie went on.
“This is his third trip, an outward trip, the fourth trip will be back to the start point and the fifth trip will be the last one, outward. This means he can go 1/3 of 1,000 miles further, or 333 miles, taking him to the 533-mile point, where he can leave 334 bananas and still have 333 bananas left to get him back to the 200-mile point, where he can pick up 200 bananas to get him back to start point. This still leaves 200 bananas at the 200-mile point. It also leaves 334 bananas at the 533-mile point. He will use these bananas for his last trip out to the oasis.”
Camel snorted, but it almost seemed an appreciative snort, which, no doubt, is what it was. It is hard to understand the emotions of camels. Tollie continued, coming to the last leg of the journey. Camel paid careful attention.
“Okay. Now he sets out on the last trip, with 1,000 bananas. He will eat 200 to get to the 200-mile point, but he will pick up the last 200 bananas there, and have 1,000 bananas again!”
Tollie paused to breath. This was, he thought, the toughest challenge he had so far faced.
“With a thousand bananas at the 200-mile point, he goes to the 533-mile point, eating 333 bananas to get there. BUT, he picks up the 334 bananas he left there on the second trip. So, he has 1,001 bananas now. He can only carry 1,000 bananas at a time, maximum load. So, he eats one banana for the first mile before he sets out. He then takes 1,000 bananas, eating one per mile.