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May 2004  

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logo puzzle of the month 1 Puzzle #97
Quiz/test #7 logo pzm 2
logo pzm 3 W-kammer #7
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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 r-percent 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?
a lot of money! francais/italiano
solution

  The problem states that any investment at r-percent 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 mid-range 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):
e0.01rn = 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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Test your knowledge online
1. Tokyo was know as:
a) Kendo
b) Kyoto
c) Edo
2. The oldest inhabited city is:
a) Rome, Italy
b) Jerusalem, Israel
c) Damascus, Syria
3. Spell 'hard water' using only 3 letters! (lower-case)
a) b) c) a) b) c) complete

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Puzzle #9, logic, by Giuseppe Spina
In the middle of a pond lies a water-lily. The water-lily 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

Wunderkammer #7
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The unfillable Klein Bottle

Klein bottle, kleinsche Flasche, bottiglia di Klein, bouteille de Klein  The Klein bottle - devised by the German mathematician Felix Klein (1849-1925) - 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 non-orientable surface that has no inside or outside (i. e. a fly moving around in this surface can return to it's starting point mirror-imaged!). It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, 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 3-D space (R3), it shows that it contains one closed, continuous curve of self-intersection, 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 2-dimensional 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
maccaroni to klein bottle
More Klein sandglass  
Book: the 4th Dimension  
Klein sandglass:  (click to enlarge)
Interesting sites: Lego® Klein bottle
A lot of Klein bottle images (in French)

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