The M1M1 treasure hunt, week 6

"Back so soon, James? I thought you'd done enough for this term."

"Urgent summons from M, Moneypenny. There's been a new release. Without me, the Sky Falls." So saying, Bond tossed his hat onto the stand in the corner and knocked on the door to the inner office.

"Come in 007, or should I say 007². There's been a serious break-in at the Foreign Office. Apparently some sort of decoder ring has gone missing. Too sensitive to bring in the police, so it's up to you."

"A decoder ring? Isn't that schoolboy stuff?"

"Not that sort of ring, 007. It's a mathematical construct of some sort. Vital to the defence of Her Majesty's Realm. They asked for my best agent. But you'll have to do."

Bond raised an eyebrow. Humour hadn't been part of the old M's repertoire. "Any clues, ma'am?"

"Apparently a small child with hairy feet was seen in the vicinity. Probably a coincidence, but it may be worth investigating. Before you go, we have a new 'Q' who has developed a toy he wants you to try out."


"Now pay attention, 007," instructed the gadget expert. "This is a particularly useful invention. It resembles a standard, hand-held mobile device, with inbuilt camera, GPS, weather and sports reports, calculator - all the usual apps. But it's actually phony. If you type in a suitable code on the numerical keypad it will enable you to talk to anyone in the world."

Bond stared in consternation at the young, spotty replacement of his old friend. "That's absolutely brilliant, Q - I don't know how you think of these things. How would I talk to you, for example?"

"My number is the value of the fifth derivative of x³e^64x/5 - x²e^4x/2 + e^x evaluated at x=0. Try it."

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"You've got the hang of it. But anyone who rings it without good reason will get an angry response. Good luck."

Bond was shown round by a stumpy government scientist, George O'Lum.
"It's gone, gone!" wailed Dr G. O'Lum. "It was so precious to me."
"Are you sure you're in the right story? This plot is getting somewhat contorted," muttered Bond.

"Ah, but Maths brings all stories together, fact or fiction. It deals with universal truths. O My Precious."

"Tell me about the child with the hairy feet," asked Bond.
"Oh him. Odd little chap. No shoes. Gave an address in The Shire. Kept asking me riddles. Things like, 'What have I got in my pockets?' and:

'A cylindrical tin can of fixed volume has radius r and height h. What is the ratio h/r which minimises the amount of metal needed to make the can including the lids?' I got as far as πr²h and 2πrh+2πr² and then he ran off, muttering something about a problem sheet."

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Car chases were more customary than calculus calculations for Bond, but neither shaken nor stirred he continued calmly "And what address did he leave?"
"Oh, that was another riddle. He lives in 'Bag End,' but the number was something cryptic. I wrote it down somewhere. Oh yes:"

"Let y(x) be given implicitly by y + cos(y) = x. What is the modulus of d²y/dx² when sin y=4/5?"

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