Re: Math puzzle: Feynman's lesson on bragging

On 17 Apr 2006 13:06:14 -0700, mensanator@xxxxxxx wrote:

jerry_friedman@xxxxxxxxx wrote:
In /Surely You're Joking, Mr. Feynman/ (pp. 194-5--thank you, Amazon
Book Search), our hero tells how once at Los Alamos he bragged that any
problem that someone could state in 10 seconds, he could solve in a
minute to within 10%. Several co-workers tried to stump him, but he
made all the estimates in time. Then another co-worker named Paul Olum
heard the challenge and unhesitatingly said, "The tangent of 10 to the
100th." As Feynman put it, he was sunk.

So my puzzle is: What would Feynman's best guess have been?
Mathematically, given a fraction x (in this case, x = 0.1), what number
y should you guess in order to have the highest probability that (1 -
x) tan(10^100) < y < (1 + x) tan(10^100)? And while you're at it,
what's the probability that your guess y will be in the allowed range?

If you can imagine calculating tan(10^100) in a reasonable amount of
time, make the exponent big enough so you can't imagine calculating the
answer within the lifetime of the universe.

(The answer is simple enough that I can imagine it might be possible to
get without calculus, but I have no idea how.)

In Python,
math.tan(10**100)
-0.59463637945948455

But Windows calculator says it's
0.40123196199081435418575434365382

I guess I still don't know how.

Mathematica indicates that Windows calculator is pretty close:

Mathematica 5.2 for Mac OS X
Copyright 1988-2005 Wolfram Research, Inc.
-- Terminal graphics initialized --

In[1]:= N[Tan[10^100],50]

N::meprec: Internal precision limit $MaxExtraPrecision = 50.
reached while evaluating Tan[100000000000000000000000000000000<<50>>00000
0000000000000].

Out[1]= ComplexInfinity

In[2]:= Block[{$MaxExtraPrecision=200},N[Tan[10^100],50]]

Out[2]= 0.40123196199081435418575434365329495832387026112924

In[3]:= Tan[N[10^100,200]]

Out[3]= 0.4012319619908143541857543436532949583238702611292440683194415381168\

71809822119121146726730974932083

In[4]:=


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.