Author's Notes

This fic was inspired between a mix of Junpei's philosophical talk and some of my own thoughts mixed with psychology, the lack of communication between these two characters in Frontier, Asarikou-chan's fic A Misaimed Question and a scientific experiment (or rather what I would call a pseudo-experiment) described by Less Wrong in the fic Harry Potter and the Methods of Rationality, chapter 8: Positive Bias, which is similar to the last question but I was following a different range of thought and the purpose was different. It reminded me of the questions in our quarterly editions of the mathematical magazines at the university. Certainly fun, but involves quite a bit of thinking. I got the second one completely wrong.

Enjoy, and tell me what you think. Also, if you could do me a favour and tell me which category you think you fall into, or suggest a new one, that would be welcome too. Or riddles, and I'll try my head at them.

Scientific Forte

And to think, they were having a perfectly intelligent conversation. Problem was, everyone else was having trouble following the train of thought.

Junpei S & Kouichi K

'No, no, no.' Five sets of eyes blinked at Junpei who was waving his hands. 'The question is: "how many characters does the right answer to this question have?" And that's English characters mind you, not Japanese.'

'What sort of question is that?' Takuya muttered, before going back to his lunch.

Junpei looked at him. 'It's called *logic* Takuya.'

'A number,' Kouichi said suddenly, before the brunette could answer back, putting his chin on a fist except for the thumb and adapting what Patamon called his "thinking pose."

Tomoki blinked at the elder twin, but Kouji nodded thoughtfully. 'I guess that makes-'

'Four,' he interrupted, before looking up and finding his brother with his mouth half-open and in the middle of a sentence. He blushed slightly, having not intended to interrupt him. 'Gomen Kouji.'

'Four?' Kouji repeated. 'Oh, I get it.'

'Great,' Izumi said, fixing her ponytail again. She had messed it up while trying to make sense of the question. 'Then explain it to those of us to whom that made no sense.'

'Four has four English characters,' the younger twin said plainly, breaking apart his chopsticks and getting a start on the rice.

'Yeah, but what makes you so sure that's the right answer?'

Kouji shrugged. He wasn't the one who had come up with the answer after all.

'It's the only number that, when spelt in English, has the same number of characters as the number it's representing,' Junpei said. 'But how'd you figure that out so fast?'

They all turned to stare at the elder twin, who luckily for him had his eyes on his own chopsticks.

'All you had to do was count the characters,' Kouichi explained. 'In one, that's three, then two, three again, and three which has five. Then comes four, which has four characters and because that's a valid answer, there's no need to count any further.'

'Oh.' He leafed through the magazine he was reading before stopping on another page. 'Have you heard the lost your way one?'

'No, I don't think so.' Four shaking heads accompanied the statement.

'It goes like this. "You're travelling to some village. At some point there is a fork in the road. You could go two ways but only one of them leads to the village. Lucky for you there are two men standing next to the fork. But unfortunately one of them always lies and one always speaks the truth and you do not know who is who. Since the men do not really like to help you, you are allowed to ask one of them only one question. Which question should you ask?"'

There was a stretch of silence.

'Um…can I ask another few questions?' Takuya offered, causing Kouji to stifle a laughter. 'Hey, you do better.'

That caught the younger twin off guard. 'Will you lie to me?' he said, after another awkward minute.

'That won't tell you which path in the fork is the right one,' Kouichi pointed out. 'It just tells you which of the men is the one who will always lie and which will always tell the truth. In any cases, both men will just say "no".'

'He is on the right track though,' Junpei said, grinning and looking up. 'Sort of.'

'I'm napping that paper,' Izumi decided, giving up trying to visualise that scenario. 'Can't you just not ask them anything and try both routes? Or ask someone else except for a bunch of unhelpful men? Or buy a map?'

'That's not a part of the question Izumi,' Junpei sighed as Tomoki giggled. 'You're supposed to ask them a question that will get you to the village.'

'So if you asked them the way, one would tell you the right way and the other would tell you the wrong way, right?' Kouichi asked. At a nod, he continued thoughtfully. 'If you asked them the wrong way, they'd just reverse. What you need is a way to ask them the way and get the same answer from both.' He thought for another minute under a few bewildered stares and then said: 'If you asked them what the other person would answer if you asked *them* which path would lead to the village, the one who tells the truth would point to the wrong one and the one who lies would also point to the wrong one. So you'd just pick the other and wind up at the village.'

'Yep,' the eldest of their group nodded. 'Okay, you other guys don't answer this one.'

'Why not?' Tomoki asked, trying to peek over Junpei's shoulder. 'I didn't even get to say anything.' He finally succeeded as Junpei continued flicking the pages. 'Oh.'

'What?' asked four curious voices.

'Not telling,' Junpei said hurriedly before Tomoki could, before turning to their youngest member. 'You're not telling either.'

Both twins took that moment to get some food into their stomachs, both having abandoned their meals in favour of thought.

'Give me a science puzzle and I can figure it out,' Kouji muttered. 'But logic has a habit of not making sense to me.'

'Okay,' Junpei said. 'Then you can answer this one. "Suppose you sit in a rubber boat in a swimming pool. In the boat lies a stone. You throw this stone from the boat into the pool. Will the water level in the pool raise or drop?"'

'Drop,' he said immediately.

'I would have thought rise,' Izumi said about two minutes later.

'Me too,' Tomoki and Takuya both agreed.

'I thought it stayed the same,' Kouichi admitted.

'You're talking about the amount of water, not the height of space it occupies,' Junpei explained, still looking for his next question. 'Actually, Kouji's right.'

Kouji rolled his eyes, specifically at Takuya. 'It's simple physics. The water doesn't weigh nearly as much as the stone.'

It still didn't make sense to anyone.

'Has that got to do with buoyancy?' Takuya asked finally.

'Hn, so you do pay attention in class.'

'Hey, I don't see you lecturing them!'

'That's because Kouichi and Izumi are both in the arts stream and Tomoki's still in junior high.'

'Okay,' Junpei sighed. 'I found the next question, and remember, no answers except for Kouichi.'

Kouichi stared at him but set down his chopsticks.

'Find a number consisting of 9 digits in which each of the digits from 1 to 9 appears only once. This number should satisfy the following requirements: The number should be divisible by 9, if the most right digit is removed, the remaining number should be divisible by 8, if then again the most right digit is removed, the remaining number should be divisible by 7. etc. until the last remaining number of one digit which should be divisible by 1.'

'That'll take forever,' Takuya said immediately, having forgotten or else abandoned his earlier discussion with Kouji.

'Taku-'

'Hey, I'm just making an observation,' he protested as the dark haired boy dug around his pockets for a paper and a pencil. Finding one of each, he set to work scribbling.

Half an hour later, he was still scribbling.

'I need a calculator,' he sighed.

'Uh…yeah, you do,' Junpei agreed, rubbing the back of his head. 'What did you get up to?'

'The fifth letter is five,' Kouichi replied, now struggling to read through his scribbles. That was a habit he really had to fix.

'How'd you figure that?' Takuya asked.

'Well, the numbers are between one and nine and any number divisible by five has to end with 5 or 0.'

'And you remember those rules?'

He blushed again. 'I'm hopeless at mental arithmetic so I thought remembering rules of the sort would make it easier.'

'Does it?' Tomoki asked.

'Not a substitute for mental arithmetic,' Kouji replied before his brother could. 'Taking the extra time to remember rules and then apply them slows you down.'

'Well, you are the one in the science stream.'

'And I have always failed to see why you can answer questions that sound utterly ridiculous and yet can't manage to compute a simple math problem. Like what's 37 multiplied by 51.'

'Aren't they both primes?' Kouichi asked, before frowning slightly in thought. Trying to do that in his head was really impossible…for him anyway. His mind didn't work that way. 'Don't know.'

'It's 1887,' Kouji replied, as everyone, even Junpei, wondered how he did it. 'But your usual rules don't work. Like the one where you can cut a number in half and times the other by two to make the multiplication easier.'

'Yeah, the problem with prime numbers is you can't split them.' He folded his paper. 'I'm afraid I don't see the point to this. Unless you can list down every number with nine digits, using each of numbers one to nine with the fifth letter as number five, the second and sixth letters one of 4 and 8, the fourth and eighth letters one of 2 and 6 and the first three, first six and first nine letters, when summed, being divisible by three.' He paused for breath, and something else occurred to him. 'Hold on, make that also the sum of letters four, five and six being divisible by three. That should be easier since we know the fifth letter is five.'

'And basically all you have to do is keep following that and you'll wind up at the solution,' Junpei nodded. 'Right.' He flicked to the last page. 'Okay. On this page is a rule that applies to sequences of three numbers. You're going to tell me the rule, and the only thing you're allowed to do is give me sequences, and the only thing I'm allowed to do is say whether or not it follows the rule, and to start you off with a set: 2,4,6.'

That got Kouji's curiosity.

'And only Kouichi,' the eldest said hurriedly. 'But the rest of you can peek.'

They did so with varying levels of apparent enthusiasm.

'And no comments,' he hurriedly added as Takuya went to open his mouth. The brunette shut it again.

'Um…1,2,3?'

'Yes,' Junpei answered.

*Okay, so nothing to do with even numbers, or going up by twos.*

'5,6,43?'

'Yes.'

*Okay, no going up by anything.*

'3,2,1?'

'Yes.'

*No going up at all.*

'-3, -5, -8?'

'Yes.'

'2, 2, 2.'

Kouichi bit his tongue lightly in thought.

'0.5, 0.7, 0.2?'

'Yes.'

'Pi, -e, ln(2)?'

'Yes.'

Another pause.

'What in the world is left?' He blinked once as the other opened his mouth. 'Hang on.'

Junpei closed his mouth again.

'0,1,2?'

'Yes.'

'All real numbers?' Kouichi asked slowly, failing to come up with anything else. 'Unless there's an imaginary plane…'

'There is,' Junpei admitted. 'But this question wasn't made for people who've studied complex numbers. That's in the final year and only in the advanced mathematics class. So the answer is actually all real numbers.'

'So if I said an imaginary answer, you would have said no, right?'

'Yep, that would be the general idea.' The magazine snapped shut.

'Okay? What exactly was the point of asking Kouichi all that?' Takuya asked. 'I can't be the only one who's completely lost.'

'To win myself a bet,' Junpei said cheerfully, and if any of them had been walking at that current point, they would probably tripped on a forgotten obstacle and lain sprawled on the ground.

'How so?' Izumi managed to ask finally. 'I'm afraid I'm having trouble visualising this?'

'Yeah, probably.' Junpei scratched his head, but didn't go on to explain, instead turning to the twins. 'How about you Kouichi?'

'Comparing my way of thinking and reasoning to someone else's?' he answered, though he made it sound like a question.

'Like who?'

'I'm not a mind-reader Junpei.'

That caused Tomoki to start giggling, and even Kouji cracked a smirk.

'All right, all right.' Junpei waved his magazine, before realising he did so and putting it away in his bookbag. 'Kouichi, you're an arts student. So is Izumi, so how come you could answer each logic puzzle and she couldn't?'

'I'm no good at logic?' Izumi offered. 'Or maybe his ability to see pictures in his head is better than mine.'

'Or maybe it's the fact that you see pictures,' the other countered. 'I noticed that in the Digital World. Anyway, Takuya mostly follows instinct, so logic doesn't work well at all, but if I asked him what to do in a social situation, or even say during the battles in the Digital World, he'd probably have the right answer, and yes Kouji, that is including that plan with Duskmon.'

'Teaching lessons,' Kouichi explained quietly.

'Yep,' Junpei nodded. 'And Kouji's pretty much the exact opposite, following a set of rules. But in a way, you're like Izumi. Seeing pictures. Like getting a whiteboard in your head and doing mental arithmetic or drawing out the electron configuration of a carbon atom.'

'You can't prove that.' But Kouji both looked and sounded rather surprised.

'No. It's probably unconscious by now, but it fits and you can't deny that.'

He couldn't.

'Kouichi on the other hand, while following rules, doesn't have that whiteboard. That's why he can grasp what seems like obscure logic but fail to work out a relatively simple arithmetic problem without a pen and paper or a calculator.'

Kouichi cocked his head slightly. 'I hadn't thought of it that way,' he said thoughtfully. 'But what would you substitute the whiteboard with?'

'Well, I was imagining something like a puzzle with more than one solution, but so far that's the best I can come up with.'

'And that disproves the polar opposite theory,' Kouji nodded. 'I should remember that.'

'And me?' Tomoki piped up, feeling rather left out.

'I'm still working on that,' Junpei said, a little hurriedly. 'You're definitely less likely to detach from the rails than Kouichi…' He stopped at the four odd looks he received. Oddly enough, Kouichi wasn't one of those four. 'I mean, if I asked him right now how to get into a house with three doors, one with poison on the handle, one with electricity and one perfectly safe but he doesn't know which one is which…' He stopped, otherwise his point would fall through. 'Kouichi? The answer?'

When the other didn't answer straight away, Junpei ploughed on. 'Tomoki?'

'Knock?' he suggested, in far less time.

'See my point?'

'Um…no.'

'But what has this got to do with your bet?' Izumi asked exasperatedly.

'Err…that's a long story.'

'Can't be any longer than this,' Takuya groaned. A lot of the time, he didn't get the point of these sorts of discussions.