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Previous Puzzles of the Month + Solutions
May 2004

 Puzzle #97 Quiz/test #7 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? 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'! Get the Archimedes Month's puzzles on your web page!

Previous puzzles of the month...

 August 98: the irritating 9-piece puzzle September 98: the impossible squarings October 98: the multi-purpose hexagon November 98: the incredible Pythagora's theorem December 98: the cunning areas January 99: less is more, a square root problem February 99: another square root problem... March 99: permutation problem... April 99: minimal dissections July 99: jigsaw puzzle August 99: logic? Schmlogic... September 99: hexagon to disc... Oct-Nov 99: curved shapes to square... Dec-Jan 00: rhombus puzzle... February 00: Cheeta tessellating puzzle... March 00: triangular differences... Apr-May 00: 3 smart discs in 1... July 00: Funny tetrahedrons... August 00: Drawned by numbers... September 00: Leonardo's puzzle... Oct-Nov 00: Syntemachion puzzle... Dec-Jan 01: how many squares... February 01: some path problems... March 01: 4D diagonal... April 01: visual proof... May 01: question of reflection... June 01: slice the square cake... July 01: every dog has 3 tails... Aug 01: closed or open... Sept 01: a cup of T... Oct 01: crank calculator... Nov 01: binary art... Dec 01-Jan 02: egyptian architecture... Feb 02: true or false... March 02: enigmatic solids... Apr 02: just numbers... May 02: labyrinthine ways... June 02: rectangle to cross... July-Aug 02: shaved or not... Sept 02: Kangaroo cutting... Oct 02: Improbable solid... Dec-Jan 03: Hands-on geometry Feb-Mar 03: Elementary my dear... Apr-May 03: Granitic thoughts June-July 03: Bagels... September 03: Larger perimeter... Oct-Nov 2003: square vs rectangle Dec-Jan 04: curvilinear shape... February 04: a special box March 04: magic 4 T's... April O4: inscribed rectangle

Quiz #7
 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

 Everyone has at least one logic or math puzzle that is his or her favorite. Send us yours and let all our readers enjoy them!

Posted puzzles
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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
 Puzzling facts

The unfillable Klein Bottle

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
 More 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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