Lexus wasn't new to the library. In fact, he'd been there many times - five times in the past week alone. But this time, he felt different stepping through the wooden door into the spacious area that was the college library.

Lexus fished his phone out of his pocket and opened the list of books that he noted down earlier. The first book on the list was "How to think like a mathematician: a companion to undergraduate mathematics" by Kevin Houston, and it seemed like a good place to start.

After picking the book out of the shelf, Lexus headed to a quiet study room and sat down on one end of a long wooden table. Skimming through the book, there didn't seem to be much content that he didn't already know. But the primary focus of the book seemed to be methods of proof and handy mathematical tools, and he couldn't go wrong with building a solid foundation of mathematics. Lexus brought out his pen and paper, and with book on one side paper on the other, Lexus quickly entered a state of flow.

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Mathematics EXP +1

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Ding! A noise went off in his head, and Lexus almost jumped from his chair. Looking up, he immediately noticed the glaring notification screen and froze.

Luckily, no one seemed to be staring at him. Breathing a sigh of relief, Lexus leaned back on his chair. If no one could see the screens or hear the system, it made things much more convenient. Why did he even bother copying down the list of books? After a moment of spacing out, Lexus looked back down and continued where he left off, 30 pages into the book.

Page after page, scribble after scribble, Lexus slowly read through the book. The occasional sound notified him of a EXP increase, but they barely distracted him from what he was reading. By the time his stomach reminded him that he really needed to eat, Lexus was onto the third section of the book and gained a couple more experience points, all in mathematics.

Before he left the library to pick up some food, Lexus wanted to see how much experience he had earned so far. Wondering whether he could communicate with the system telepathically, Lexus tried calling out to it in his head.

Nothing.

Lexus shrugged. What did he expect?

"System, can you show me my status screen?" Lexus whispered, careful not to let anyone else overhear. A glowing screen appeared before his eyes. Not much changed, other than a single line:

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Mathematics: Level 0 [EXP 4/100 (4%)]

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Lexus was pretty excited. If he managed to get 4 experience points by reading two sections, then it should be possible to level up with just this quest alone. Level 1 wasn't that unreachable after all!

The desire to gain more experience and level up meant that Lexus wasted no time with his lunch. He gorged down the food served at the college dining hall and returned straight to the library to continue with reading. He had two hours before his badminton session, which was enough time to get midway through section three.

For Lexus, the day was pretty much completely filled with reading "How to Think Like a Mathematician". He went to badminton, headed straight back to the library and continued with the book until it was time for dinner. The library didn't close until midnight, so he stayed until the library closed, borrowed the book, went back to his room and continued reading.

However, reading math textbooks was always a slow and time consuming task. It was past midnight, three days later, and he was at his desk reading with the study lamp switched on when he turned the final page.

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Congratulations on completing book: How to Think Like a Mathematician

Quest progress: 1/10

Mathematics EXP +5, Computer Science EXP +5

---

Lexus closed the book and thought back to his experience in the past few days. He had been driven by the desire to gain experience and level up, but he also found the book itself really interesting. It was a book that focused more on mathematical technique rather than teaching new content, and Lexus was able to see a lot of the work he was doing in a new light. Instead of having to grapple with the problem sheet mathematical monsters in the dark, it felt to him like he had a guide that pointed out the monster's weaknesses and suggested plans of action.

But of course, what made him feel the most excited at that moment was still the sweet, sweet experience points. "System, status!"

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Name: Lexus Kagan

Current Quest: Background Knowledge [1/10]

Computer Science: Level 0 [EXP 5/100 (5%)]

Mathematics: Level 0 [EXP 17/100 (17%)]

Physics: Level 0 [EXP 0/100 (0%)]

Engineering: Level 0 [EXP 0/100 (0%)]

Chemistry: Level 0 [EXP 0/100 (0%)]

Biology: Level 0 [EXP 0/100 (0%)]

---

Lexus couldn't hold back his smile when he saw the two digit number appear on his status screen. He was making much better progress than he originally anticipated. After all, 17 was a heck of a lot better than 5. At this rate, he would only require another five books before his mathematics level increased from 0 to 1.

Looking at the book list, he decided to prioritise the mathematics textbooks over the computer science ones. He was almost certain that the natural experience gain was rewarded based on the contents of the book and the kind of knowledge he was learning, so focusing on mathematics textbooks would allow him to level up as quickly as possible.

Lexus decided that his next five books would be as follows:

1. Ross, S.M. (2014). A First course in probability

2. Hefferon, J. (2009). Linear Algebra

3. Biggs, N.L. (2002). Discrete mathematics

4. Machler, M., Solan, E., and Zamir, S. (2013). Game Theory

5. Lehman, E., Leighton, F.T. and Meyer, A.R. (2014). Mathematics for computer science

With some luck, he would be able to finally gain enough mathematics experience points to level up. Onwards!

=====

"A First Course in Probability" was nothing like "How to think like a mathematician", not even close. It wasn't so hard that it was physically impossible to read, but it was definitely on a whole other level. Out of curiosity, he went online to read the reviews, and pretty quickly gathered that the name of the book, "A First Course in Probability", was a complete lie.

But that wasn't a reason to give up. Lexus loved a challenge. What better feeling could there be than the sense of exhilaration when you've managed to accomplish something thought to be difficult?

20. Balls are randomly removed from an urn initially containing 20 red and 10 blue balls.

What is the probability that all of the red balls are removed before all of the blue ones have been removed?

The first thought that came to Lexus' mind as he laid his eyes on the question was "I should try smaller examples". He had learnt from experience that when you didn't know how to approach a question, trying smaller examples would always help.

For example, simplifying the question from 20 red balls and 10 blue balls into 2 red balls and 1 blue ball.

Lexus gathered that the easiest way to get the solution for 3 balls would simply be to list all the possible orderings. He wrote down "RRB", "RBR", "BRR", then paused.

There was no other solution that he could find, so the answer to the question in the simplified case was a third. Which sort of made sense.

But how would he generalise that solution to the case with 30 balls? Lexus couldn't possibly list all of the possible outcomes.

If the final answer was a third... it must have something to do with the proportion of blue balls to the total number of balls.

Looking at the simplified case again, Lexus studied the three solutions closer.

Oh! Oh. Ohhhhhh.

The condition was satisfied as long as the last ball removed from the urn was a blue ball. So by ordering all 30 balls and finding out how many of those had a blue ball as the last ball, then the answer could be calculated easily.

The total number of orderings is 30!⁄20!10!=30045015, and fixing the last ball to be blue, the number of orderings for the remaining 29 balls would be 29!⁄20!9!=10015005. The answer would therefore be 10015005⁄30045015, which simplifies to 1⁄3. Nice!

As the thoughts rushed into his mind, Lexus transcribed the working faithfully onto a piece of paper. The moment he finished writing down the full solution to the question, a familiar yet unexpected screen popped up.

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Mathematics EXP +3

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It wasn't the first time he gained experience from reading, but it was definitely the first where he gained more than 1 experience point at a time. A full 3 experience points! Under normal circumstances, he would have to read for a few hours even think about gaining that many points.

Lexus paused to think about his strategy going forward. It seemed like the system rewarded him whenever his mind 'clicked', like if he discovered something he didn't know before or if he made a connection between seemingly distant concepts. In order to maximise the number of these experiences in the future, Lexus decided that he would skip the easier exercise problems and focus on problems that looked more difficult.

To test this theory, Lexus flipped through the rest of the book and picked a question that he thought looked interesting:

17. Three points X1, X2, X3 are selected at random on a line L. What is the probability that X2 lies between X1 and X3?

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-one hour later-

Ummmm... where to start?

At this point, Lexus had no choice but to admit that he got ahead of himself. He hadn't been able to get anywhere with the question, and he also hadn't gained a single point of experience. No ding, no notification screen, no 'click' in the brain, no nothing. Just endless, black, abyss. Multiple pieces of paper were filled with scribbles in an attempt to shed some light, but none of them had led him anywhere even close to the solution.

"System, can you give me a hint?"

Nothing.

"Please?"

Still nothing.

"System, status!"

Finally, a response:

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Name: Lexus Kagan

Current Quest: Background Knowledge [1/10]

Computer Science: Level 0 [EXP 5/100 (5%)]

Mathematics: Level 0 [EXP 21/100 (21%)]

Physics: Level 0 [EXP 0/100 (0%)]

Engineering: Level 0 [EXP 0/100 (0%)]

Chemistry: Level 0 [EXP 0/100 (0%)]

Biology: Level 0 [EXP 0/100 (0%)]

---

Good, he hadn't been hallucinating for the past two days. Just checking.

Lexus was determined to solve the question, but he knew full well that he could only do so by learning the prerequisites and arming himself with the right knowledge and tools. In resignation, Lexus flipped back to where he previously was in the book, and continued reading with pen and paper in hand.

The best strategy was to read slightly above his level, i.e. not to let himself get too comfortable, but not to be complacent and jump forward to areas he didn't even begin to understand. Lexus stuck to first reading through each chapter in the book, then attempting the first few questions in the "Problems" section at the end of the chapter just to warm up. He would then skim through each following question in succession until he found one that he didn't know how to solve just by looking at it. At this point, he would give himself half an hour to attempt to solve it.

Most of the time, Lexus managed to find the answer, or at least make some good progress on the question by the time the half-hour was up. But in the rare occasions where he didn't make any meaningful progress, Lexus took a picture of that question and sent it to Arthur. A typical conversation would go like:

Lexus:

An urn contains n white and m black balls. The balls are withdrawn one at a time until only those of the same colour are left. Show that with probability n/(n + m), they are all white.

Lexus: "Arthur, I was stuck on this question, could you look through it and give me some hints?"

Arthur: "Imagine the possible sequences of withdrawals of balls from the urn. What kind of properties would a sequence have if all of the remaining balls were of the same colour?"

Lexus: "I don't know, I guess if they have r balls left then the last r elements of the sequence would be the same?"

Arthur: "And? Does it matter what the value of r is?"

Lexus: "Ummmm... No, because if r is smaller, you would just remove more balls..."

"Oh I get it! As long as the last ball is white, you can keep on withdrawing balls until there are only white ones left. So it's just equivalent to the probability that the last ball is white in a given sequence?"

Arthur: "Yep"

From that point on, Lexus was able to work out the solution on his own. The probability of the last ball being white is simply the proportion of white balls to the total number of balls, which is n/(n+m) as expected.

It was at this moment that Lexus realised what an idiot he was. He had done this question before, just in a different form. You know how he got the probability of 1⁄3 on the question with red and blue balls? Guess what's 10/(10+20)? Yeah.

Ding!

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Mathematics EXP +1

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Great, the system was mocking him again.