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# Missing Square Puzzle!!

What if it was real?!

Note by Vasavi GS
2 years, 10 months ago

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Infinite chocolate! Time for Willy Wonkers to research into it :) Staff · 2 years, 10 months ago

The cutting line is not a complete straight line.
missing square = the difference between the actual straight line and the cutting line in this problem · 2 years, 10 months ago

Indeed. And a way to see this mathematically, is that the 'cutting line' consists of 3 points, namely (0, 2.5), (3, 3.5) and (5, 4.5). But this is not a straight line (compare the gradients), and merely appears straight. In fact, it has area equal to

$\left| \begin{matrix} 0 & 3 & 5 & 0 \\ 2.5 & 3.5 & 4.5 & 2.5 \\ \end{matrix} \right| = 1$

(The above is known as the shoelace formula. You can calculate it the old fashioned way if you are unfamiliar with this.)

This accounts for the additional piece of chocolate.

YUM! Staff · 2 years, 10 months ago

It seems so. · 2 years, 10 months ago

right... · 2 years, 10 months ago

If you will see with full concentration at it you will found that the pieces of the block containing 3 rectangles has base which will gradually increases as it moves up you can see it by focusing it's a nice trick · 2 years, 10 months ago

Yeah! You're right! Great observation. · 2 years, 10 months ago

The way it was cut diagonally resulted in a gap so small you can't visibly see which amounts to the "left over piece". · 2 years, 10 months ago

I think if we analyze by taking screenshot of each section we can find that the shape at the edges is not correct after they connect .They changing the shape while connecting it again.

I have taken screenshot and attached in the following link.U can check the same and analyze. http://www.ltewirelesstech.com/2014/03/just-see-carfully-and-analyze.html · 2 years, 10 months ago

never thought of this wooow.. · 2 years, 10 months ago

I have a feeling that the diagonal line is being cut in such a way that $$\frac{1}{5}$$ of each small piece/cell of chocolate involved in the cut is being added again, thereby making the whole chocolate cell adding process quite inconspicuous.

I'm probably wrong though. · 2 years, 10 months ago

That's actually very close to the truth! It's not a diagonal line that is being cut out, but more of a thin triangle. Staff · 2 years, 10 months ago

This is merely a corollary to the Tarski-Banach Theorem, of course there's nothing funny going on here. If one can decompose a sphere and make 2 spheres, each the same size as the original, from it, surely we can make an extra chocolate piece here. · 2 years, 10 months ago

Not quite. There is something much more basic going on here.

Seeing is not believing. Staff · 2 years, 10 months ago

Calvin, can't you see a joke? Bringing in the Tarski-Banach Theorem is like bringing in a flame thrower to kill a fly. Others here have already correctly identified the problem, kudos to these folks. · 2 years, 10 months ago

Sorry, irony / sarcasm rarely works well when communicated over the internet.

What do you think is the best way of explaining exactly how the Banach Tarski paradox works? Staff · 2 years, 10 months ago

"Exactly how" is the problem. This is an exceedingly difficult theorem to follow or explain on the internet, so the best that can be said for the layman is that it is mathematically possible to decompose a solid sphere into a finite number of pieces, which can then be rearranged and reassembled to form TWO solid spheres identical to the first! But in practice, this is impossible to carry out because the pieces are "non-measurable", i.e., do not have a well-defined volume. Think of Mandelbrot fractals. Imagine that a piece would resemble one, an infinity of point clusters fixed relative to each other so that it is a "piece', but no one can actually create such a physical piece. It's a mathematical abstraction. How is it possible, then, to make 2 somethings out of 1? Well, again, it's related to the fact that half of infinity is still infinity. It is as mind boggling as the fact that 1 + 2 + 3 + 4 + ... adds up to -1/12, but nevertheless mathematically consistent.

This theorem does depend on the "Axiom of Choice", though, it presupposes that given any infinity of non-empty sets, it's always possible to create a set which contains exactly one element from each of those sets. This might seem obviously true, but, like the parallel postulate, nobody's been able to prove this. · 2 years, 10 months ago

its nothing but a illusion that if go no to a series we come to first row last second piece (which is cutted half) while sliding(moving up) it increases in its size. I HOPE YOU WILL GET IT. · 2 years, 10 months ago

Actually the size decreases! · 2 years, 10 months ago

or so they say. · 2 years, 10 months ago

But how? Ossama Ismail above also gave a possible way it decreases but still I haven't got it in my head · 2 years, 10 months ago

my Brain si Bleeding... · 2 years, 10 months ago

The part which is cut from right and fitted into left, will not fit exactly because that is less than what was cut from left but it is depicted so in the animation and same thing for right side. and hence the additional piece of chocolate will not be there in reality · 2 years, 10 months ago

it is like the shifting of origin of perfect square · 2 years, 10 months ago

Wow.......Can it be really happen?? · 2 years, 10 months ago

how can I post a picture????? · 2 years, 10 months ago

There's a 'attach image' (look here)option when you share a problem or note. · 2 years, 10 months ago

no in the comment box · 2 years, 10 months ago

It has many choclate or uncountable choclate!!!!!! · 2 years, 10 months ago

if you see with concentration you will find out that it is not possible, as size of pieces are not equal... · 2 years, 10 months ago