Do the math! (NYC) Mystery Cache
mmacgown: This cache has been difficult to maintain and since it gets few visitors, I'm letting it go.
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Difficulty:
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Terrain:
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Size: 
(small)
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Cache is NOT at the listed coordinates. You must solve the puzzle provided below in order to find the necessary coordinates. Note that in order to solve this puzzle, you need to be able to view the file linked to the main description page (i.e., a .gpx file likely won't cut it). You are ultimately looking for a small container, into which the smallest of trackables may fit. As always, BYOP!
One of the things that is pretty neat about Manhattan is all the puzzle caches concentrated here. There are a lot of clever folks that have put a lot of thought/love into crafting some fantastic caches along these lines. While I still have only stumbled upon the tip of the iceberg in terms of solving these puzzles, this cache represents what I hope to be a fun contribution.
While this cache does involve a degree of elegance (associated with the mathematical concepts), this 'puzzle' requires more of a brute force attack. To find the necessary coordinates, you must solve three undergraduate freshman calculus-level mathematics problems. Specifically, you will need to solve a definite integral, perform some complex (pun!) arithmetic, and finally solve a nonlinear ODE and apply a set of boundary conditions. Note that while some of these parts could be solved numerically (i.e., on a calculator or computer), they might not be solved with the appropriate precision so to ensure that the required numbers are obtained. Thus, I suggest finding the solutions analytically (and subsequently rounding off to the required numbers of significant figures as noted) so to minimize headaches.
[contact me if you'd like a higher-res pdf version]
Clearly, this puzzle will be a delight to those who love math. And unfortunately a dread to those who don't like it (or find these problems way too easy!). If you are stuck, I'd suggest teaming up with someone (e.g., roommate, spouse who remembers math from college, child who is in AP calculus right now, etc...) who can help you with figuring out the underlying mathematical concepts (but not simply do the problems for you!). Hopefully everyone who finds this cache will have taken a route that provides some degree of mathematical joy.
As a bonus, I have three mathematical texts from my personal library, one for each of the FTF, STF and THF (and only for those who independently solved the puzzle; group finds will have to share their collective prize). Finders will be contacted about coordinating a means of transfer.
Have fun with this and good luck!
Bee Bait
PS Feel free to contact me if you get stumped. Am happy to try to help (I am a math teacher...)
[6.22.11] Congrats to childofatom for taking the 'phone a friend' path to the FTF. Big congrats to Heyseanmcbride for taking the direct path there.Your prize on statistical analysis is coming its way. Now, who will claim the other two prizes?
Additional Hints
(Decrypt)
Fgntr 1n: Hfr gur punva ehyr va erirefr (v.r., vagrtengvba ol cnegf). Ohg bayl bapr. Qbvat vg gjvpr jvyy yrnq lbh va pvepyrf....
Fgntr 1o: Pbairegvat gb cbyne pbbeqvangrf znxrf guvf n fvzcyr trbzrgel ceboyrz...
Fgntr 1p: Ybtvfgvpnyyl, guvf ceboyrz vf rnfl gb fbyir vs lbh frcnengr inevnoyrf naq cresbez n cnegvny senpgvba qrpbzcbfvgvba. Vagrtengr naq fbyir sbe nal haxabja pbafgnagf.
Fgntr 2: Abgr gur greenva engvat (naq or pnershy jura vg vf jrg bhgfvqr!). Sne pbeare (gbjneqf TJO), nccebkvzngryl purfg urvtug.