### Brain Teasers

# Area of a Square

You have a piece of paper, 10cm by 10cm. Area = 100cm^2. For some reason, you need a square piece of paper with an area of 50cm^2. Using the paper you have, what's an easy way of getting the new square?

### Hint

Try it with a piece of paper, doesn't matter the size of the square, try to find a square half its size inside it.### Answer

Fold the four corners of the square into the centre. This doubles the thickness of the paper, and so halves the area.Hide Hint Show Hint Hide Answer Show Answer

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## Comments

It's a really good teaser!

But it's hard for me. Sort of!

But it's great!

But it's hard for me. Sort of!

But it's great!

You could also just fold the 100x100 square it in half and then half the other way to give a 50x50 square.

Gd answr though!! I would never have thought about doing it like that!

=¬)

Gd answr though!! I would never have thought about doing it like that!

=¬)

If you fold it in half, then half again to make a square, it's 5cm * 5cm = area 25cm^2, so it doesn't fit the answer.

Brilliant. Absolutely, completely, snark-farking brilliant. I love it.

Who writes these questions so poorly? Here's another mis-worded one. First you want a 50cm^2 area, and then you want a square. Other problems or riddles here lack sufficient information allowing multiple correct answers. Straingle, not Braingle!

Also, what's with this time-out feature that lets me ponder a teaser for only a minute and then I have to refresh my e-mail to get to the site for answers? What's the point?

you're right, it should have been worded better, stating the paper had to remain a square earlier, thanks for pointing it out.

I understand english and thought the riddle was too hard

I understand english and thought the riddle was too easy i mean... i don't know how you thought it was hard.. i said rip it into 1/4s though

I'm not trying 2 b meen or anything, but personally, i thought that it was a little dumb.

the question stated that the solution had to be a square.Does it matter where in the question it said so? I liked this one. For deanlilja, I ponder questions for ages with no problems. I suggest the problem could be with your pc or modem. do you have a short idle setting?

I had no idea that you cound fold the corners of the square like that and end up with sides of 7.0710678118654752440084436210485 cm each.

Hey, The teaser is pretty neat, and the idea was everybody saying "Dumb, it is easy, just folded half and half" later to find out that that will yield to a 25 square.

Great teaser.

Great teaser.

Great teaser!

There is another way to do so (not that smarts though). Simple math shows that diagonal of the new square is 10cm long. So fold original square so that its diagonal is aligned with one of its 10cm long sides. This corner will show the end of the new 10cm long diagonal. Now it’s easy to make new square.

I'm not sure why there's so much confusion on this teaser. Seems pretty obvious to me that you need to make a square of 50 cm^2.

Very nice. It was so simple, but I couldn't get it. Awesome.

I was gonna say cut it

It is really wierd!! i didn't get it!! I don't think it's that great!

WHY NOT JUST FOLD IN HALF

10X10

10/2=5

5X10 = 50

OTHER SIDE STAYS THE SAME

10X10

10/2=5

5X10 = 50

OTHER SIDE STAYS THE SAME

you can just fold it in half...........

Has this teaser been edited? Do some people not understand the word square? I don't really think the given solution provides a single piece of square paper. I agree with the principal but my solution was to join all 4 midpoints of the original square and cut off the triangular corners formed. (Very handy bit of knowledge is that the length of the diagonal in any square is root 2 times the length of the side.)

Good puzzle

Good puzzle

gosh!!, I can't believe that I actually got it right, I did some algebra first and found out that the length of one side of a square whose area is 50 cm^2 should be 5 square root of 2, I immediately knew that this was half the length of the original square's diagonal, and then, I put another diagonal for each of the four "quadrants" of the square and saw that it was the simply same length as half of the original square's diagonal, the next events were accidental, I didn't even know that I was already forming a square!!! THIS PUZZLE IS COOL!!!

Another neat way of doing it:

Inscribe a circle in the square. then inscribe a new square in that circle. The smaller square has half the area of the larger one

Inscribe a circle in the square. then inscribe a new square in that circle. The smaller square has half the area of the larger one

cooperkid: You only have the paper, no compass and straightedge.

Well, maybe cooperkid is the greatest freehand drafter in the world. Just kidding.

Love this teaser though, brilliant!

Love this teaser though, brilliant!

I'm with coolkid. Where in the question does it state that I do not have access to a ruler and compass?

There is nothing in the question that says you can not use other instruments to do it.

Of course folding the paper is an easier way to do it.

There is nothing in the question that says you can not use other instruments to do it.

Of course folding the paper is an easier way to do it.

I have a doctorate in math science, so this was extremely easy.

HABS: Explicit: "You have a piece of paper. ... Using the paper you have..."

Implicit: "You don't have anything other than the paper. You may not use anything other than the paper."

Implicit: "You don't have anything other than the paper. You may not use anything other than the paper."

With these kind of teasers I just read them and then go on to the comments. You have to be a mathematician and that I am not.Never have been good with anything pertaining to figures. Loved all the comments though, and the different opinions. That was fun!!

I am also with those who do not think that the framing of the puzzle restricts the solver to using only the original piece of paper, since the instructions do not say, "Using ONLY the piece of paper that you have...", and if they did say that explicitly, then would one be allowed to pick the piece of paper up using, say, one's fingers, which would be using something other than the original piece of paper itself? In any event, this is an interesting teaser, and the proposed solution does, in essence, the same thing as the solution given by those who would simply inscribe into the 100 Sq. cm square a smaller square, formed by connecting the midpoints of each of the sides, N to E, E to S, S to W, and W to N. By the way, doubling the thickness of the paper is entirely irrelevant, as one can easily understand when one considers that simply folding the original square in half would also double the thickness of the sheet, and would produce a parallelogram of half the area of the original square, but that parallelogram wouldn't be a square.

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