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Puzzle
#97 

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Suppose
you've placed a 'certain' amount of money
in a secret savings account in Switzerland
at rpercent interest... To know
roughly when your capital will be doubled,
you can use the trick of Luca
Pacioli, an Italian monk and mathematician
of the Rinascimento: simply divide the
'magic number' 72 by the
interest rate r and you'll obtain
the number of years you should wait for...
Could you explain why and how it works? 


The
problem states that any investment at rpercent
interest per annum will be doubled in
approximately [72 / r] years.
a)
Using simple compound interests
Since the initial principal is to be doubled we have the equation
(formula for
compound interests):
2 = (1 + r/100)^{n}
Taking the log of
both sides:
log 2 = n · log(1
+ r/100)
or
n = log 2 / [log(100 + r)
 2]
Next
we seek to find a number, x, such that
when divided by r, the result
is approximately n.
Thus
x / r = log 2 / [log(100 + r)
 2]
Then
x ~ 0.301r / [log(100 + r)
 2]
Nowadays
interest rates are comprised between:
0.5 <= r <= 7
Selecting a midrange value r = 3,
we find for x
x ~ 0.301 · 3 /
(log103 
2) ~ 70.35
b)
Using continuously compound interests
Since the initial principal is to be doubled we have the equation
(formula for
continuously compound interests):
e^{0.01rn} =
2
Taking ln on
both sides:
0.01rn · ln e =
ln 2
or
n =
ln 2 / 0.01r
Next
we seek to find a number, x, such that
when divided by r, the result
is approximately n.
Thus
x / r = ln 2 / 0.01r
Then
x ~ ln 2 / 0.01 ~ 69.31
The
above proves that Pacioli found an 'acceptable'
duplication value Formula for compound
interests; nevertheless, 70 (or
even 71) would be a
more accurate 'magic number'!

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Quiz
#7 

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Puzzle
#9, logic, by Giuseppe
Spina 
In
the middle of a pond lies a waterlily.
The waterlily doubles in size every day.
After exactly 20 days the pond will be
completely covered by the lily. How many
days were necessary to cover half of the
pond? 
Rate: •• 
Solution
#9 

Puzzle
#10, maths, by K.
Z. Urpania 
1
liter French wine + its bottle cost 1'000
dollars. The wine costs 998 dollars more
than the bottle. Find the cost of each! 
Rate: •••• 
Solution
#10 




The
unfillable Klein Bottle
The Klein
bottle  devised by the German
mathematician Felix Klein (18491925)
 was originally called a 'Kleinsche
Fläche' [Klein surface]
in German but mistranslated into English
as 'Kleinsche Flasche' [Klein bottle].
Well, a Klein bottle is a closed nonorientable
surface that has no inside or outside
(i. e. a fly moving around
in this surface can return to it's
starting point mirrorimaged!). It
can be constructed by gluing both
pairs of opposite edges of a rectangle
together giving one pair a halftwist,
but can be physically realized only
in 4 dimensions, since it must pass
through itself without the presence
of a hole. The image opposite is an
'immersion' of the Klein bottle in
a 3D space (R3), it shows that it
contains one closed, continuous curve
of selfintersection,
where the 'neck' meets the 'bottle'
(it is where the light green part of
the image meets the dark green part).
This topological object is like a Moebius band without
edges! Actually, it is what you get when you glue 2 Moebius bands along their
edges. Of course, as said before, you need an extra space dimension to do this...
So now, you know why a Klein bottle cannot be filled, because there
is no hole at all in it. But how can we visualize a real Klein bottle without
holes? We can often visualize abstract concepts like this if we shift down one
dimension. Let's think about a 2dimensional equivalent: suppose we live on a
plane, and want to draw a simple closed curve that looks like a figure eight
(8). The trouble is that it intersects itself,
so it's not a simple closed curve. But if we had a good enough imagination to
picture a third dimension, we could lift one part of the curve a little in that
direction, making a 'bridge' so the figure eight didn't intersect! That's the
sort of thing you have to do to make the Klein bottle work (without holes)...
A question, what happens if we make a 'real' hole in a Klein bottle? We transform
this topological surface into another one called Moebius
slip.
We'll end this article by a joking quotation: "in the topologic
Hell the beer is packed in Klein bottles!".

Transformation
of a macaroni into a Klein bottle


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ORIGINAL Wunderkammer fact 



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•••
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Robert Beauvais
•••
Math Gems
logx = 0.4343
lnx
logarithms' identity
••• 


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