M1M1 Treasure Hunt

The M1M1 treasure hunt, weeks 5-6

Chapter 3: The Matryoshka Matrix

"Morning!" breezed Ann, as she eagerly entered the blue box once more.
"Who are you?" asked the baffled Doctor slumped behind his desk.
Ann was momentarily nonplussed, but swiftly took control.
"You may call me Ann. I'm a time traveller. I usually confine myself to moving into the future at about 600000 seconds per week. It's only natural for you to find this disorientating at first, but you'll get used to it. You haven't met me yet, but you will a few weeks ago and we got on well. Oh, and amazingly, my room is the same size inside as out, allowing for the thickness of the walls."
"Are you sure we got on? I didn't find you at all disrespectful and flippant?"
"No, why would you?" Ann smiled disarmingly. "You promised to take me to a formative time in the development of our lecturer's interest in Algebra."
"I did? I mean, I will? I suppose I better had then, as I might already have done." And so saying he began to fiddle with the controls.

While waiting for the outcome, Ann asked "Why did you travel two weeks into the future anyway?"
"It's complicated - to do with College politics. I believe there's another time machine somewhere, owned by my evil Nemesis. You haven't noticed another blue box, much bigger than this one, anywhere around have you?"
"I'm afraid not. Well, apart from The Blue Cube on the walkway, of course."
"What? Where? I knew it! Is it an exact cube? I bet it isn't - I know these embodiments of evil. It'll be a cuboid, with width w, depth d and height h. Three of the vertical walls will be covered in solar panels, disguised as dark blue windows. The total surface area of these panels will be fixed at 4h, in suitable units. Given this, w and d will have been chosen to give the maximum possible total volume, which is
. ?

But I'll worry about that later," the Doctor concluded. He flipped the switch and the lights flickered.
Ann opened the door and beheld a city street at dusk, with snow swirling around a lit window. Through a gap in the curtains a young boy could be seen playing with a bookcase. Dr Hu and Ann moved closer to see what he was doing.

The boy couldn't quite reach the top shelf of a bookcase and so he left it alone. He then systematically rearranged the 2nd row of books leaving a gap on the far left. Having dealt with that shelf to his satisfaction, he started on the one below, this time leaving a still larger gap. And so he continued his operations, until the array of books had a distinctively triangular appearance. With an enquiring air and surprising strength, he lifted up the bottom left corner of the bookcase and rolled it over until the top row of books was now at the bottom. He seemed oblivious to the resultant crashing to the floor of a small samovar, a collection of self-similar, hollow wooden dolls, a vodka bottle and other stereotypical paraphernalia.

"Непослушный мальчик!" screamed a voice from next door, and an adult figure ran in.
"What does that mean?" whispered Ann.
"Oh don't worry - the automatic translation feature will soon cut in - it's an essential plot device," the Doctor reassured her. Sure enough, after a slight delay, the boy burst into tears and cried
"I only wanted to see if the array was invertible!"
"Well you put all the books back as they were right now, Alexei, or it's straight to bed with you," his parent scolded, rashly replacing the dolls, samovar and surprisingly unbroken bottle on top of the inverted case.
The boy examined the books, most of which were now the wrong way up, and considered the best way of righting them all. With a delighted smile, he realised all his previous steps were reversible and reached for the bottom right corner of the case....

"Yes, I can see that would have been a formative experience," mused Ann, as they retreated from the loud and unhappy after-math.
So what are you doing in lectures now?" asked the Doctor once they were out of the cold.
"Differentiation. It's so easy. For example, the 6th derivative of x3e2x evaluated at x = 0 is

." ?

"Bah! Everything's either too easy or too hard. Try this. Can you think of a function which is equal to its 4th derivative? Of course you can. There are millions of them: cos(x+1), sin(x), cosh(x), 2sinh(x), 2ex, -5e-x...so choose any 4 of them; make sure they're linearly independent, and write them in the top row of a 4 x 4 matrix."
"OK, I've done that. Am I going to do some row operations like the young Professor out there?"
"Sort of. You're going to differentiate the first row, and write the answers in the 2nd row. Then you're going to differentiate the 2nd row and write the answer in the 3rd row. Finally differentiate the 3rd row and write the answers in the 4th row. You get it? So each column consists of one of those functions and its first 3 derivatives."
"Done that too. But I'm unhappy about where this is heading - I'm not even going to try to invert this matrix," said Ann.
"Nothing so major. You just have to take the determinant of the 4 x 4 matrix. And then differentiate the answer. While you're working it out, I'll take us home."