When i was young i read an article in Scientific American about how a
square can be divided into different shapes, which are then combined
into a new shape, which in turn would result in an increase in size.
Someone argued that they could use that to increase volume and hence
the mass of an object. which they would use to create extra gold
this of course was proven incorrect.
the question is, can you locate that article, and if not, can you
locate the appropriate resources about the problem and how to prove
its incorrectness?

Dear pi314159265,

You are thinking of a Mathematical Recreations column by Martin Gardner
that I have read myself. I can look up the exact volume and issue numbers
at my university tomorrow, if you like, but it won't help you unless you
also have access to decades-old back issues of Scientific American. The
column is copyrighted material, and SciAm's online archives extend back
only as far as 1991. Gardner's column appeared much earlier than that.

As an alternative, you may wish to order a book that explains the secret
behind this puzzle and many of its variants. In Gardner's Mathematics,
Magic and Mystery, Chapters 7 and 8 are devoted to puzzles of this
kind. The book is published by Dover and sold by Amazon at a modest price.

Dover Publications: Table of Contents for Mathematics, Magic and Mystery
http://web.doverpublications.com/cgi-bin/toc.pl/0486203352

Amazon: Mathematics, Magic and Mystery (Cards, Coins, and Other Magic)
by Martin Gardner
http://www.amazon.com/exec/obidos/ASIN/0486203352/qid=1113134800/sr=2-1/ref=pd_bbs_b_2_1/104-6123063-2132734

Although this general class of puzzle is known as a geometrical vanish 
or a vanishing puzzle, the form in which many of us first encounter
it, as you did, is the increasing puzzle. Here, a square of 8x8 = 64
units is transformed by cutting and rearrangement into a 5x13 = 65-unit
rectangle. The increasing puzzle in reverse is, of course, a vanishing
puzzle. The following page features an interactive Java animation of
the classic increasing puzzle.

Cut the Knot: A Faulty Dissection
http://www.cut-the-knot.com/Curriculum/Fallacies/FibonacciCheat.shtml

More coverage is provided here.

University of Surrey: Ron Knott: A Fibonacci Jigsaw puzzle or How to
Prove 64=65
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles2.html#jigsaw1

Another page illustrates with great clarity that the extra square's worth
of space is accounted for by a narrow gap running along the diagonal of
the rectangle.

Jim Loy: The Extra Square
http://www.jimloy.com/puzz/missing.htm

Interestingly, the 8x8 dissection can also be rearranged into a shape
whose area appears to be only 63 units in area.

Cut the Knot: Sam Loyd's Son's Dissection
http://www.cut-the-knot.com/Curriculum/Fallacies/SamLoydSon.shtml

Finally, you may be amused by another vanishing puzzle in which a triangle
appears to lose one square unit's worth of area after rearrangement. Do
you see where the missing square has gone? It took me quite a while to
work this one out.

Cut the Knot: How Can This Be True?
http://www.cut-the-knot.com/ctk/BeTrue.gif

It has been a pleasure to address this question on your behalf. If 
you have any concerns about my answer, please advise me through a
Clarification Request so that I may attend to your needs before you
assign a rating.

Regards,

leapinglizard

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Request for Answer Clarification by pi314159265-ga on 10 Apr 2005 09:08 PDT

It would be of a great help if you can look up the issue number.

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Clarification of Answer by leapinglizard-ga on 10 Apr 2005 09:13 PDT

I shall do so tomorrow.

leapinglizard

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Clarification of Answer by leapinglizard-ga on 12 Apr 2005 19:56 PDT

In the May 1961 issue of Scientific American, one of the puzzles in
Martin Gardner's column is a geometrical vanish in which both the
original and the rearranged shape have the outline of a square, but
one of them has a hole in the middle. I'm certain there are other
Martin Gardner articles featuring such puzzles, given that they are a
recurring theme in his books. Could the May 1961 issue be the one?

If not, can you give me some idea of when you read the article, say
within a range of five or ten years? It would help a great deal with
the search. By the way, I have taken a look at the book chapters I
recommend above and they contain an awful lot of information about
these kinds of puzzles. If you're interested in the mathematics behind
them, I'm certain you'll be glad to own the book.

leapinglizard

I know the trick your referring to... its an illusion where you
rearrage triagles within a triangle and form another triangle and you
seemingly get less area... its a trick.  You can not rearrage a fixed
area into another area.  Anyone who shows you something to the
opposite is just providing an illusion.

I found a site with something similar to the problem...

http://www.jimloy.com/puzz/missing.htm

With regards to the increase in math... conservation of mass prevents
creation of new mass in this particular situation.  If someone claimed
this they were most likely trying to sell something LOL

Here is an url where you can buy all of Martin Gardner's books on a
searchable CD for about $50:

https://enterprise.maa.org/ecomtpro/Timssnet/products/TNT_products.cfm?primary_id=TDG&secondary_id=null&Subsystem=ORD&action=long&format=table

That's an interesting CD, but it's far from a complete edition of
Martin Gardner's writings. It is merely a collection of his
Mathematical Recreations columns, constituting a fraction of his
lifetime output. He has also written books of popular science and on
debunking pseudoscience, as well as annotations of Alice in
Wonderland, The Hunting of the Snark, and the Rime of the Ancient
Mariner. See the following bibliography.

David Schoen: A Martin Gardner bibliography
http://www.loyalty.org/~schoen/gardner-booklist.html

leapinglizard