A new puzzle: Multiplying bread and fish

[ark]

First and foremost notice this quote from the Bible:

34 As he went ashore he saw a great throng, and he had compassion on them, because they were like sheep without a shepherd; and he began to teach them many things.
35 And when it grew late, his disciples came to him and said, "This is a lonely place, and the hour is now late;
36 send them away, to go into the country and villages round about and buy themselves something to eat."
37 But he answered them, "You give them something to eat." And they said to him, "Shall we go and buy two hundred denarii worth of bread, and give it to them to eat?"
38 And he said to them, "How many loaves have you? Go and see." And when they had found out, they said, "Five, and two fish."
39 Then he commanded them all to sit down by companies upon the green grass.
40 So they sat down in groups, by hundreds and by fifties.
41 And taking the five loaves and the two fish he looked up to heaven, and blessed, and broke the loaves, and gave them to the disciples to set before the people; and he divided the two fish among them all.
42 And they all ate and were satisfied.
43 And they took up twelve baskets full of broken pieces and of the fish.
44 And those who ate the loaves were five thousand men.

Then notice that there is a celebrated (among the mathematicians) Banach-Tarski Paradox see for instance::

http://www.kuro5hin.org/story/2003/5/23/134430/275

Layman's Guide to the Banach-Tarski Paradox

The Banach-Tarski Paradox is well-known among mathematicians, particularly among set theorists.1 The paradox states that it is possible to take a solid sphere (a "ball"), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere. In other words, you've doubled the volume of the original sphere.

"Impossible!" I hear you say. "That violates physical laws!" Well, that is what many mathematicians said when they first heard this paradox. But I'd like to point out in this article why this may not be as impossible as one might think at first.
....

Now, look at this picture:

Can you repeat Jesus' miracle using the method indicated in the picture? Can you do it with scissors and paper, not only in your mind?

 

[aurora]

http://209.85.135.104/search?q=cache:UP0l_SIc-iIJ:wolfweb.unr.edu/homepage/alex/coll/btarnotes.pdf+solution+to+the+Banach-Tarski+Paradox&hl=tr&ct=clnk&cd=22&gl=tr&client=firefox-a

There are equations and some explanation for the above paradox, it is a bit long (which I did not understand anything)

http://www.daviddarling.info/works/Mathematics/mathematics_samples.html

Banach-Tarski paradox

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.
-William Shakespeare

A seemingly bizarre and outrageous claim that it is possible to take a ball, break into a number of pieces and then reassemble those pieces to make two identical copies of the ball. The claim can be made even stronger: it is possible to decompose a ball the size of a marble and then reassemble the pieces to make another ball the size of the Earth, or, indeed, the size of the known universe!

Before writing off Messieurs Banach and Tarski as being either very bad mathematicians or very good practical jokers, it's important to understand that this is not a claim about what can actually be done with a real ball, a sharp knife, and some dabs of glue. Nor is there any chance of some entrepreneur being able to slice up a gold ingot and assemble in its place two new ones like the original. The Banach-Tarski Paradox tells us nothing new about the physics of the world around us but a great deal about how "volume," "space," and other familiar-sounding things can assume unfamiliar guises in the strange abstract world of mathematics.

Stefan Banach and Alfred Tarski announced their startling conclusion in 1924, having built on earlier work by Felix Hausdorff who proved that it's possible to chop up the unit interval (the line segment from 0 to 1) into countably many pieces, slide these bits around, and fit them together to make an interval of length 2. The Banach-Tarski paradox, which mathematicians often refer to as the Banach-Tarski decomposition because it's really a proof not a paradox, highlights the fact that among the infinite set of points that make up a mathematical ball, the concept of volume and of measure can't be defined for all possible subsets. What this boils down to is that quantities that can be measured in any familiar sense are not necessarily preserved when a ball is broken down into subsets and then those subsets reassembled in a different way using just translations (slides) and rotations (turns). These unmeasurable subsets are extremely complex, lacking reasonable boundaries and volume in the ordinary sense, and thus are not attainable in the real world of matter and energy. In any case, the Banach-Tarski paradox doesn't give a prescription for how to produce the subsets: it only proves their existence and that there must be at least five of them to produce a second copy of the original ball. The fact that the Banach-Tarski paradox depends on the axiom of choice (AC), yet is so strongly counterintuitive, has been used by some mathematics to suggest that AC must be wrong; however, the benefits of adopting AC are so great that such black sheep of the mathematical family as the paradox are generally tolerated.

 

[Rich]

Just had fun trying it with paper, it is perplexing. How is it possible?

 

[vinny]

Just had fun trying it with paper, it is perplexing. How is it possible?

ahaaa..!! ;) look carefully at the hypotenuse (the long side) of the triangle for a clue.

 

[Approaching Infinity]

It's a trick! It's all in the lines. More perimeter = more surface area. As for the Jesus... well, he was Jesus.

 

[aurora]

yes, Sleepyvinny, What about the hypotenuse ? the two sides of the right angle have the same number of blocks in each triangle, so the hypotenuse must be the same length for the two triangles. Or I am I missing something ?

 

[vinny]

yes, Sleepyvinny, What about the hypotenuse ? the two sides of the right angle have the same number of blocks in each triangle, so the hypotenuse must be the same length for the two triangles. Or I am I missing something ?

yep, the hypotenuse is the same length for the two triangles, but you are missing something!
if you look along the hypotenuse - you might be able to see that it is not a straight line...

 

[Beau]

Neither of the hypotenuse's are straight...

 

[Keit]

yep, the hypotenuse is the same length for the two triangles, but you are missing something!
if you look along the hypotenuse - you might be able to see that it is not a straight line...

You mean that the incline is different?

Take a look at the "meeting point" of the "red" and "yellow" pieces on the hypotenuse of the first image.
Now take a look at the exact point in the second image. It seems as if the incline is slightly different.

 

[CarpeDiem]

yes, it's like hypotenuse on the first image is very slightly curved in and on the second - very slightly curved out.

every element of SO has a pair of diametrically opposed fixed points on S

And the difference is evident if looking on hypotenyse of the 9-th horisontal quadrant numbing them from the left. Is that the trick?

 

[domi]

Can you repeat Jesus' miracle using the method indicated in the picture? Can you do it with scissors and paper, not only in your mind?

As far as where the hole came from: 5 * 3 = 8 * 2 - 1

Large triangle...not!

 

[Nević Nenad]

Noo, it is not in the hypotenuses; imagine some old man authoritive voice :)

I think it’s in not so precisely draw lines.Imprecisely draw lines spare space in quadrates angles, especially in dark green and red.

 

[vinny]

Noo, it is not in the hypotenuses; imagine some old man authoritive voice :)

I think it’s in not so precisely draw lines.Imprecisely draw lines spare space in quadrates angles, especially in dark green and red.

nope ;)
even if the lines are drawn precisely, the hypotenuse on the first one is 'concave' and on the second is 'convex' - the points where the red and green triangles meet are actually inside (on the first) and outside (on the second) of the actual hypotenuse, which you 'll have to imagine, as it isn't drawn.

so that means that the second triangle (stretching a point here, it's not actually a triangle) is fatter than the first, and the extra area of that 1x1 square is spread along the extra 'fatness' between the concave and convex versions. if you drew the two figures superimposed on one another, then you'd see it straight away.

 

[Lee.T]

Great puzzle, sleepyvinny is correct that the angles for the hypotenuse are slightly different

How about this One

3 People go to dinner, each one has a 10 pound note only
The dinner was great and now time for the bill
The bill comes to exactly 25 Pounds.
They all hand over there 10 pound note.
The waiter comes back with 5 one pound coins.
So each person takes one pound which leaves 2 pounds for the tip.
So each person came with 10 and leaves with 1 pound having spent 9 pounds.
and the waiter has 2 pounds
3x9=27+2=29
where has the other pound gone?

 

[OCKHAM]

Very interesting analysis Royalteel. I seem to agree with you. The original drawing looks as though it has something wrong even to the eye. I am not sure if you are correct though.

In your example at the restaurant, each person has paid 9.33333333…

If the bill is 25 and each added 1 to the tip that would be 28, not 27.