Tom ambled into Hermione's study the Sunday afternoon before school started. He propped himself on his elbows and looked curiously at the shining numbers floating above the parchment. "What's that?" he asked, pointing to a shape that looked vaguely like a seashell.
"That's the golden spiral," she told him bemusedly.
Tom blinked, then said, "I mean, Hermione, I know you teach Arithmancy, and that has to do with maths, and I was wondering if, you know, if you had the time to teach me something like that too…"
"How about this," said Hermione. "Since you're not going to elementary school, I'll teach you some basic maths, and we can get you books on all kinds of subjects, all right?"
Tom grinned excitedly. "Okay."
"You know the basic binary operations, right? Addition, subtraction, multiplication, and division?"
Tom nodded. Hermione remembered the first fascinating thing she had learned in maths.
"Here's an interesting problem." She reached over for a new piece of parchment, and wrote:
1+2+3+4+5+6+7+8+9+...+97+98+99+100
"The ellipsis here signifies that the sequence of positive integers goes on, all the way to 100- but writing all of them would take too much time."
Tom stared. "That's a lot of numbers to add."
"Yes, one hundred numbers, to be exact. Let me tell you a story about this problem. I first heard it from my fourth-grade teacher, and I don't know if it's true." She shrugged embarrassedly, then continued. "You know Isaac Newton, right?"
"The famous physicist, who wrote The Mathematical Principles of Natural Philosophy. The Philosophiæ Naturalis Principia Mathematica, often shortened to just the Principia. Right?" Tom smirked.
Hermione laughed. "Well, you do know your books."
"I've never read it. I was afraid to," Tom grinned.
"Well, the story goes like this. When Isaac Newton was little… maybe around your age, he was at school. His teacher had some grading to do, or just didn't feel like teaching that day, or something. So she gave this problem to her class." She tapped the numerical expression with her finger. "She figured, that huge problem, right, would take that class at least an hour to do. But little Isaac Newton had the correct answer in two minutes."
Tom was astonished. "How?"
"You see, Tom, he figured out a way to do it, much cleverly than just adding each number up. He found a way around the problem, instead of going at it head-on. Here." She handed him the parchment and a quill. "I'll give you five minutes to figure out the correct answer. That's three more minutes than Isaac Newton took!"
Tom was a little anxious. "Umm…"
"Don't worry, this is a pretty hard problem to start out with. I didn't get it until my teacher told me. But you're the smartest kid I've ever met."
"Thanks," said Tom, genuinely pleased.
"OK. Now go!"
A/N: The answer is quite simple. If you imagine, in your mind, writing out all the numbers in a row: 1+2+3+...+99+100, then adding the first and the last numbers (1 and 100), the second and second-to-last numbers (2 and 99), the third and third-to-last (3 and 98), you will see that there is a pattern. All the number pairs add up to 101. There are 100 numbers, thus there are 50 pairs. 50 times 101 is 5050, so the sum of the first 100 positive integers is 5050.