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Math Brain Teaser – Extra Space

January 12th, 2010 by Steven Pomeroy | Filed under Brain Teasers, Math Tricks.




I came across a very interesting problem – one that seems to involve math tricks of some sort.  Or is it magic?  Whatever it is, it is certainly a math brain teaser!  Anyway, here is the graphic that had me scratching my head:

So what’s the deal here??  Where did the hole come from?  Please post your answers  :)

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14 Responses to “Math Brain Teaser – Extra Space”

  1. reydel astorga | 14/01/10

    these are not triangles. if u all check carefully the hypotenuse of the figures, it is not actually a straight line. one can check on the first figure, consider the points (0,0), (8,3) and (13,5), by solving for the slope, they are not equal which should have been if they are collinear.so as the second figure by considering the points (0,0), (5,2) and (13,5).so rearranging them will not yield with the same area. i hope i did not confuse you

  2. Sam | 19/01/10

    Additional to what Mr. Reydel explained above.
    We will try to solve it in both ways, geometrically & arithmetically.

    1st method:
    – Print the Picture of these 2 “triangles”
    – Use a fine pencil and a perfect ruler and reproduce “accurately” a straight hypotenuse for each of the 2 figures.
    – You will notice that it’s not matching with the picture. In figure 1 there is a lost area and in figure 2 there is access of area. The sum of these areas will be equal to 1 square.
    And that’s where this “additional” square came from

    2nd method:
    Since this mathematical trick is based on “area” illusion, let us calculate the area of the 2 triangles arithmetically:
    Let us recall the formula of calculating the area of a triangle:
    Triangle Area = ½(b×h)
    Whereby b = base & h = vertical height
    In our case b=13 squares (the horizontal side of the triangles)
    h=5 squares (the vertical side of the triangles)
    => ½(13×5)= ½65=32.5

    the triangles area should be 32.5 squares. But if we double check by counting the squares we will find that:
    Figure 1 area = 32 square (less than 0.5 from the exact triangle shape)
    Figure 2 area = 33 square (more than 0.5 from the exact triangle shape)
    That’s equal to 1 full square. :D

    Have a nice day

  3. mina makram | 27/02/10

    whats da answer plsssssssssssssssssssss from the author

  4. admin | 27/02/10

    This is actually an optical illusion. The hypotenuse is very slightly curved, such that it covers enough extra area to account for the “hole”.

  5. Sanjaya | 1/03/10

    The Dark Green and the Red Triangles’’ angles are different hence the two big “triangles” are not really Triangles. But for the naked eye the variation in gradient is not visible on the picture.

  6. basma emad | 1/03/10

    the 2 tiangle arent equal

  7. Albert Santos | 9/03/10

    there are 13 sqaures fully occupied by the length of the red and the green figures on the first triangle. On the 2nd triangle there is only 12 squares oocupied by the dark green, yellow, and green figures. That is how you got the HOLE.

  8. albert rico | 27/07/10

    When the red triangle was transferred at the top portion, it occupied a portion of its previous location (at the mid part of the hypotenuse). That is why when rearranged, there is one square unoccupied… It was reoccupied by the red triangle.

  9. Magic Math « Acehrecsam’s Blog | 1/12/10

    […] cara kedua: menggunakan langkah arithmatika dalam menghitung luas daerah segitiga dengan: L = 1/2(axt) L = 1/2(13×5), jadi luas sebenarnya segitiga tersebut adalah 32.5, maka seharusnya jumlah square dalam “segitigga” itu kalau kita hitung harus 32.5, tapi kalau anda melihat secara detil jumlah square dari “segitiga” itu kedua-duanya, jumlahnya adalah 1……32 (kurang 0.5) dari bentuk segitiga yang tepat 2……33 (lebih 0.5) dari bentuk segitiga yang tepat dan jumlah lebih dan kurang itu adalah 0.5 + 0.5 = 1 square penuh, sehingga, “segitiga” kedua kurang 1 square penuh karena tidak sesuai dengan ukuran bentuk segitiga yang tepat. note: sebenarnya kedua2 gambar tsb bukan disebut segitiga…… baca lebih mendalam di http://mathtricks.org/math-tricks/math-brain-teaser-extra-space/ […]

  10. idk | 8/11/11

    tell me the answer idk what it is simply an illusion

  11. NAUSHAD | 9/03/12

    ½(13×5)= ½65=32.5 extera 0.05 (pic—–1)
    (13×5)= 65=33.00 extera 0.05 (pic—–2)

    0.5+0.5= 1 ans.

  12. Shayz | 10/04/12

    The slopes of the hypotenuse for both the red and green triangles are slightly different, causing the hypotenuse of the first triangle to bend in, and the hypotenuse of the second triangle to bend out.

    The slope of the green triangle is 2/5 and the slope of the red triangle is 3/8

    To test this, put a straightedge up to the picture from the top right corner to the bottom left, and you will see how the hypotenuse is ever so slightly bent at the junction of these two different slopes

  13. kshitij | 16/10/13

    the red n blue triangle are not equal

  14. Nima Sahba | 17/07/14

    In pic 1 you have 4 shapes and in pic 2 you see the same 4 shapes with different arrangement, the hole isnt counted as you could place the same 4 shapes anywhere. The illusion comes when you see the 4 shapes in pic 1, as 1 shape. And the hole doesnt belong to any of the 4 shapes.

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