M1M1 Treasure Hunt

The M1M1 treasure hunt, week 7


The remnants of the Fellowship of the Ring stared across the ravaged plains of Gondor towards the smoking fires of Mount Doom.
"So let's get this straight," grumbled Gimlibec the dwarf. "Purely as a diversionary tactic, we're going to attack Mordor, heavily outnumbered. In short, we're all going to die."
"Maybe not - I figure it this way," replied Logicless the Elf, whose tall good looks could not conceal the failures of his M1S-spent youth. "Each of us, individually, has a small chance of surviving, say 1/N. So if N of us go, at least one of us is sure to live, which means we've won! To be absolutely certain, we'll send N log(4) troops."

Granted the elf's preposterous assumption that each soldier's fate is independent of the others', and taking the limit of large N, what is the actual chance of at least one of the N log(4) troops surviving, written as a decimal?


?

"Well that's not too bad," the dwarf continued, slightly mollified. "And we're absolutely sure the Dark Lord has no idea what we're really up to?" he asked Aragorn, the once and future king.
"It's a sort of double bluff. He knows that we're not stupid, so the last thing we'd do is send our most powerful weapon unprotected into the heart of his realm."
"You mean, we actually ARE being stupid?"
"Worry not! The ways of the Wise are wondrous!" interjected Gandalf. "While weak, wretched waverers wonder woodenly whether we will win, we wizards wage widespread war well, with wholesome wisdom."
"Why? Well, whatever. I wonder where the wingbearer is now," snorted the dwarf.

Many miles away, Frodo and Sam were hacking their way through the harsh, barren lands.
"Are you OK, Mr Frodo?" asked Sam.
"I see it in my mind all the time now, Sam - a ring of fire. It taunts me continually. Just now it asked:"

"My shape is (x-a)2+(y-b)2 = c2. What is the value of |[1+(y')2]3/2/y''|? Put me on your finger and the answer to this and much more will be obvious..."

?

"See, Mr Frodo!" cried Sam excitedly, we've almost reached this crack of doom thing. Now let's get rid of this accursed ring and go home.

"I have come," intoned Frodo portentiously. "But I do not choose now to do what I came to do. The Ring is mine!"
So saying, the hobbit put on the ring at the zenith of its power. Instantly he became aware not only of abstract equations, but also of their myriad, fascinating, real-world applications. For example, he recognised that the lowly quadratic

y = xT - gx2(1+T2)/(2V2)

represented the path of a projectile launched with speed V from the origin at gradient T subject to an acceleration g. As an amusing exercise, he quickly calculated the maximum x-value which could be obtained by varying T when y = -5, g = 10 and V = 24:

?

Having solved the mathematical problem, the hobbit naturally looked around for a pebble to toss to verify the theory. He knew what inclination T to choose and 24m/s was a plausible launch velocity.
Unable to find a pebble, Frodo instead pitched a small golden ring he happened to find in his hand. It sailed through the air in slow motion towards the bubbling lava pit, 5 metres below....


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