(It said almost everything you needed to know about Hermione Granger that she had never let that stop her, or even let it interfere with her love of being tested.)
"The sad thing is," said the boy, "you probably did everything the book told you to do. You made a prediction that would distinguish between the robe being charmed and not charmed, and you tested it, and rejected the null hypothesis that the robe was not charmed. But unless you read the very, very best sort of books, they won't quite teach you how to do science properly. Well enough to really get the right answer, I mean, and not just churn out another publication like Dad always complains about. So let me try to explain - without giving away the answer - what you did wrong this time, and I'll give you another chance."
She was starting to resent the boy's oh-so-superior tone when he was just another eleven-year-old like her, but that was secondary to finding out what she'd done wrong. "All right."
The boy's expression grew more intense. "This is a game based on a famous experiment called the 2-4-6 task, and this is how it works. I have a rule - known to me, but not to you - which fits some triplets of three numbers, but not others. 2-4-6 is one example of a triplet which fits the rule. In fact... let me write down the rule, just so you know it's a fixed rule, and fold it up and give it to you. Please don't look, since I infer from earlier that you can read upside-down."
The boy said "paper" and "mechanical pencil" to his pouch, and she shut her eyes tightly while he wrote.
"There," said the boy, and he was holding a tightly folded piece of paper. "Put this in your pocket," and she did.
"Now the way this game works," said the boy, "is that you give me a triplet of three numbers, and I'll tell you 'Yes' if the three numbers are an instance of the rule, and 'No' if they're not. I am Nature, the rule is one of my laws, and you are investigating me. You already know that 2-4-6 gets a 'Yes'. When you've performed all the further experimental tests you want - asked me as many triplets as you feel necessary - you stop and guess the rule, and then you can unfold the sheet of paper and see how you did. Do you understand the game?"
"Of course I do," said Hermione.
"Go."
"4-6-8" said Hermione.
"Yes," said the boy.
"10-12-14", said Hermione.
"Yes," said the boy.
Hermione tried to cast her mind a little further afield, since it seemed like she'd already done all the testing she needed, and yet it couldn't be that easy, could it?
"1-3-5."
"Yes."
"Minus 3, minus 1, plus 1."
"Yes."
Hermione couldn't think of anything else to do. "The rule is that the numbers have to increase by two each time."
"Now suppose I tell you," said the boy, "that this test is harder than it looks, and that only 20% of grownups get it right."
Hermione frowned. What had she missed? Then, suddenly, she thought of a test she still needed to do.
"2-5-8!" she said triumphantly.
"Yes."
"10-20-30!"
"Yes."
"The real answer is that the numbers have to go up by the same amount each time. It doesn't have to be 2."
"Very well," said the boy, "take the paper out and see how you did."
Hermione took the paper out of her pocket and unfolded it.
Three real numbers in increasing order, lowest to highest.
Hermione's jaw dropped. She had the distinct feeling of something terribly unfair having been done to her, that the boy was a dirty rotten cheating liar, but when she cast her mind back she couldn't think of any wrong responses that he'd given.
"What you've just discovered is called 'positive bias'," said the boy. "You had a rule in your mind, and you kept on thinking of triplets that should make the rule say 'Yes'. But you didn't try to test any triplets that should make the rule say 'No'. In fact you didn't get a single 'No', so 'any three numbers' could have just as easily been the rule. It's sort of like how people imagine experiments that could confirm their hypotheses instead of trying to imagine experiments that could falsify them - that's not quite exactly the same mistake but it's close. You have to learn to look on the negative side of things, stare into the darkness. When this experiment is performed, only 20% of grownups get the answer right. And many of the others invent fantastically complicated hypotheses and put great confidence in their wrong answers since they've done so many experiments and everything came out like they expected."
"Now," said the boy, "do you want to take another shot at the original problem?"
His eyes were quite intent now, as though this were the real test.
Hermione shut her eyes and tried to concentrate. She was sweating underneath her robes. She had an odd feeling that this was the hardest she'd ever been asked to think on a test or maybe even the first time she'd ever been asked to think on a test.
What other experiment could she do? She had a Chocolate Frog, could she try to rub some of that on the robes and see if it vanished? But that still didn't seem like the kind of twisty negative thinking the boy was asking for. Like she was still asking for a 'Yes' if the Chocolate Frog stain disappeared, rather than asking for a 'No'.
So... on her hypothesis... when should the pop... not vanish?
"I have an experiment to do," Hermione said. "I want to pour some pop on the floor, and see if it doesn't vanish. Do you have some paper towels in your pouch, so I can mop up the spill if this doesn't work?"
"I have napkins," said the boy. His face still looked neutral.
Hermione took the can, and poured a small bit of pop onto the floor.
A few seconds later, it vanished.
Then the realisation hit her and she felt like kicking herself. "Of course! You gave me that can! It's not the robe that's enchanted, it was the pop all along!"
The boy stood up and bowed to her solemnly. He was grinning widely now. "Then... may I help you with your research, Hermione Granger?"
"I, ah..." Hermione was still feeling the rush of euphoria, but she wasn't quite sure about how to answer that.
They were interrupted by a weak, tentative, faint, rather reluctant knocking at the door.
The boy turned and looked out the window, and said, "I'm not wearing my scarf, so can you get that?"
It was at this point that Hermione realised why the boy - no, the Boy-Who-Lived, Harry Potter - had been wearing the scarf over his head in the first place, and felt a little silly for not realising it earlier. It was actually sort of odd, since she would have thought Harry Potter would proudly display himself to the world; and the thought occurred to her that he might actually be shyer than he seemed.
When Hermione pulled the door open, she was greeted by a trembling young boy who looked exactly like he knocked.
"Excuse me," said the boy in a tiny voice, "I'm Neville Longbottom. I'm looking for my pet toad, I, I can't seem to find it anywhere on this carriage... have you seen my toad?"
"No," Hermione said, and then her helpfulness kicked in full throttle. "Have you checked all the other compartments?"
"Yes," whispered the boy.
"Then we'll just have to check all the other carriages," Hermione said briskly. "I'll help you. My name is Hermione Granger, by the way."
The boy looked like he might faint with gratitude.
"Hold on," came the voice of the other boy - Harry Potter. "I'm not sure that's the best way to do it."