Hooper's paradox

Geometry Level 2

An $8 \times 8$ square with area 64 is split into 4 pieces, as shown on the left. Then, the 4 pieces are rearranged to form a $13\times 5$ rectangle whose area is 65.

How did this happen?

Some of the shapes in the rectangle are overlapping in a way that creates the illusion The shapes in the square don't fit up exactly as shown in the rectangle, and their areas differ by 1 The individual shapes on the left are not the same as those on the right It must be a work of the Illuminati

3 solutions

Marta Reece
Jun 16, 2017

You can see the missing area in the center.

Sorry, Marta, your diagram is not accurate. You have a 5 X 12 rectangle instead of a 5 X 13 rectangle. But, as I always say, I'm only exaggerating for the sake of accuracy!

Guiseppi Butel - 3 years, 11 months ago

My bad. Fixed it. Thank you for pointing it out.

And I absolutely love your saying. I think I'll steal it sometime, but only with your permission.

Marta Reece - 3 years, 11 months ago

You're welcome to use it. I don't remember the origin so I won't take credit for it.

Guiseppi Butel - 3 years, 11 months ago

Could i ask why option c is not acceptable?

Sheryl-Lynn Tan - 3 years, 11 months ago

I believe that option C is acceptable, and have actually reported the problem for that reason, but somehow the report was "resolved" without the option being either eliminated from the lineup or changed.

Marta Reece - 3 years, 11 months ago

Guiseppi Butel
Jun 19, 2017

The slope of the hypotenuse of the triangle is .375 and that of the slant side of the trapezoid is .4, therefore the "diagonal" is not a straight line. Therefore there will be a "gap"....

Www Www
Jun 16, 2017

From https://www.math.uni-bielefeld.de/~sillke/PUZZLES/jigsaw-paradox.html:

One of the oldest theorems about Fibonacci numbers, due to the French astronomer Jean-Dominique Cassini in 1680, is the identity Fn+1 × Fn-1 - Fn × Fn = (-1)n. The shortest proof of this identity is via the matrix identity which is another way of writing the recurrence formula. Then use the theorem that the determinant is multiplicative. n [ 1 1 ] [ Fn+1 Fn ] n | Fn+1 Fn | [ 1 0 ] = [ Fn Fn-1 ] => (-1) = | Fn Fn-1 | There is a geometric reasoning if we compare the two rectangles Fn+1 × Fn-1 and Fn × Fn. They both contain the rectangle Fn × Fn-1. This means

Fn+1 × Fn-1 = Fn × Fn-1 + Fn-1 × Fn-1 and Fn × Fn = Fn × Fn-1 + Fn × Fn-2. Their difference is Fn+1 × Fn-1 - Fn × Fn = - (Fn × Fn-2 - Fn-1 × Fn-1). Showing the relation for n=1 with F2 = F1 = 1 and F0 = 0 completes the proof.

@Calvin Lin However, as you can see, the image is misleading. It makes two right angled triangles on each other while really, the triangle-like shapes have a big gap in the middle, which is very visible

Www Www - 3 years, 11 months ago

@Calvin Lin - Due to complaints about the 'segments aren't the same ' answer, can you please change it to 'It's a miscount, and the arithmetic doesn't work properly'

Www Www - 3 years, 11 months ago

Unfortunately, this kind of problem is hard to wrangle, to make the phrasing unambiguous and have a unique answer. I've edited the problem + options, but I'm still not fully satisfied.

Calvin Lin Staff - 3 years, 11 months ago

Isn't "It is an (optical) illusion" also correct?

Calvin Lin Staff - 3 years, 11 months ago

@Calvin Lin Not really. As you can see from Marta Reece's answer, the real 'rectangle' has a gap in the middle, and the pictures are misleaders

Www Www - 3 years, 11 months ago

Right, so in that sense, it is an optical illusion because the width is so small, and the image size is so small, that we do not notice the gap.

Calvin Lin Staff - 3 years, 11 months ago

Hooper's paradox

3 solutions

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