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Where did a black square go?

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Paste a sheet of graph paper on a piece of cardboard. Draw the square shown in figure 1, then cut along the lines to make five pieces. When you rearrange these same five pieces, in the manner shown in Figure 2, a hole will appear in the center of the square!

The square in Figure 1 is made up of 49 smaller square. The square in Figure 2 has only 48 small squares. Which small square has vanished and where did it go?

civonamzuk  May 1, 2015

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https://www.desmos.com/calculator/vlhv7vswdq

fiora  May 2, 2015
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The "square" on the left doesn't have an area of 49; its area is only 48 1/2!

Box 1 has an area of 7.

Box 2 has an area of 16.

Box 3 has an area of 16.5.

Box 4 has an area of 6.

Box 5 has an area of 3.

Total = 48.5.

When arranged on the left, it's stretched to look like 49.

When arranged on the right, it's shrunk to look like 48.

Neat problem!

geno3141  May 2, 2015
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No,the total area of box1,box2,box3,box4 and box5 in the figue 1 are equal 49.

When compare each sides of box2 and box3 in figure 1 and figure 2,you find out that its different.

Now.let,s assume that 2 and 7 are 2 legs of box 1 (right triangle),and the length of  bottom side of box 3 are equal 3 (both figures) and prendicular to the left side of box 3 (both figures).

So the left side of box 3 in figure 2 is equal to 5,and the left side of box 3 in figure 1 is equal to

6-3*(2/7)=6-6/7=36/7.and other sides of box 3 and box 2 are different ,too.

fiora  May 2, 2015
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https://www.desmos.com/calculator/vlhv7vswdq

fiora  May 2, 2015
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That was great.

MathsGod1  May 2, 2015
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Yep, a small difference changes everything.

In box two at the top there are 4 cut squares.

but in figure 2  there are only 3 cut square in box 2.

MathsGod1  May 2, 2015
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This really is a fabulous puzzle !!

The area is definitely 49units squared.  I think Fiora was saying the same thing

It is the diagonal line at the top that causes the deception :)

I used coordinate geometry to find the the y value when x=4 and when x=3  This is the outcome.

You really do lose a litttle 1 unit square in the middle.

That is because the height has been extended by a skinny rectangle of area 1u^2 is

Melody  May 2, 2015
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I have added this to our Puzzle Sticky topic

Melody  May 2, 2015
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Figure  2

This image clearly shows you what happened after the peaces were rearranged.

civonamzuk  May 2, 2015
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Hey Civonamzuk

Aren't you impressed with my answer?

Melody  May 3, 2015
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Great answer Melody, I didn't even notice that box.

Sweet grid work!!!

MathsGod1  May 4, 2015
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Thanks MG

Melody  May 4, 2015
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To be honest, Melody, your answer is half right, and here's why:

The distortion of the large square and 7 small squares happened at the top, not at the bottom.

In the Figure 2, parts 2 and 3 have pushed part 1 upward, and the large square became the rectangle;

7 small squares in the top 2 rows became distorted by taking in that missing small square.

As a result, they also became the rectangels.

Please, take a look at my not-so-fancy post, Figure 2, and I'm sure you'll understand my points.

civonamzuk  May 4, 2015
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Hi Civonamzuk,

My answer is completely right - so is yours

It can be 'pushed' upwards OR downwards - makes no difference.

I had not noticed that you had even answered the question until you pointed it our just now :)

Melody  May 5, 2015

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