The Puzzle

Luke was a clever young boy; most discerning adults could see that. Seeing as he was Layton's apprentice and all, he'd already solved quite a few puzzles that would render most thrice his age snivelling wrecks of their former selves. More than that, though, Luke loved to learn: his love of puzzles was matched only by his inquisitiveness (and maybe his love for animals). Whatever he was told, he'd absorb it into his head pretty quickly. It was a bit of a stretch to call him a genius – but clever was certainly the right word for him. As far as he was concerned, Professor Layton was the real, bonafide genius, although Luke did like the sound of 'clever' for himself.

This cleverness persisted, even as Luke got on in years. He went to school (!) and was considered one of the brighter pupils in class. However, since Luke almost never seemed to use what he learned at school in puzzle-solving or in anything that was not related to school at all, for that matter, he had to wonder what point there was in his education. Being Layton's apprentice was so much more educational and useful than some crummy old institution, he thought. (Thus, he chose to ignore the fact that the professor himself was a learned man who had gone to university, among other things.)

The matter came to a head one year when Luke was sixteen and he was taught the epitome of useless things: calculus.

The horrid, vile thing! Luke couldn't bear to even look at his homework and thus kept it perpetually shoved away in the back corner of his desk. He would have set it on fire as well but that wasn't what a gentleman did. It wasn't that Luke didn't understand calculus from the teacher's explanation, but it just didn't seem usable to him at all. Frankly, he had much better things to do with his time.

Layton ended up hearing of this the way he always seemed to hear about Luke's problems without ever actively prying. He mentioned it as the two of them had tea together (Luke still liked Oasis Berry; his tastes had never evolved) and had just finished solving a word puzzle in the newspaper.

"I see you have been having quite the trouble with your homework lately, Luke," Layton remarked amiably.

Just the sheer mention of it was enough to wipe away Luke's smile and replace it with a dark scowl.

"It's not trouble, it's simply awfully boring, Professor!"

"Nevertheless," Layton reminded him, "disregarding one's obligations is not a gentlemanly thing to do." Noticing Luke's scowl still hadn't gone away, Layton added, "Now, Luke, why don't you try showing me your homework?"

"Show you, Professor?" Now Luke was confused.

To this, the good professor simply smiled and spread his arms invitingly in that comforting way he always did. "Think of it as a puzzle! Does that make it more manageable?"

"A puzzle..." Luke repeated slowly. He scratched his head. "I... I'll try...!"

Once he had retrieved his homework, he showed it to Layton. Layton read through the sheet quickly enough, and then a smile came upon his face as he neared the end.

"See here," he said, pointing. "This question seems an awful lot like a puzzle, doesn't it?"

Luke read the words aloud: "The sum of two positive integers is 108. Find the numbers such that their product is a maximum." He hesitated. "Er..."

"There is a beauty in mathematics," Layton said, "such that its entire foundations are grounded entirely upon logic and reason. These are precisely the same things we use with puzzles. Maths can be obfuscating, but the logic is nonetheless always present. For instance, there is a way to solve this problem without using any calculus at all."

"You mean...!" Luke was amazed. "You know the answer to the question already?"

"I do," Layton replied. "Now let's see if you can't find the answer yourself."

Author's note: Try to solve this puzzle along with Luke! The next three chapters are hints to the puzzle, and chapter 5 contains the solution. See if you find the answer yourself without looking at too many hints. Read the question carefully. Here it is again in bold:

The sum of two positive integers is 108. Find the numbers such that their product is a maximum.

You can probably hear the Layton puzzle theme ringing in your ears as you read this, can't you?