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Videos / Mathematics / Missing Square







Missing Square

Missing Square
Rearrange pieces of a triangle and a square goes missing. Is it magic or math?


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Rating: 2.9 / 5 (837 votes)
Posted by nova on August 9, 2007
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Comments

Posted by John Q. Public on August 9, 2007 at 11:08 am
This is an old puzzle. The "trick" is that the outer figure does not form a real triangle.

Look at the two smaller triangles. The sides of the larger triangle are 3 x 8. The sides of the smaller triangle are 2 x 5. So the triangles are not actually similar. (That is to say, the blue is not really a reduced version of the red... their angles are different.) The long side of the outer figure, then, is not really a straight line. It bends ever so slightly where the two triangles touch. This is also true of the bottom figure when rearranged.

So while the outer figures look like two triangles at first glance, they are actually 4-sided because of the slight bend.

Because the two smaller triangles do not have the same internal angles, when rearranged the other pieces fit together differently. This creates the illusion of gaining a space.
Posted by kimmy on August 9, 2007 at 8:31 pm
This is like the old joke about 3 people renting a room.
The clerk said $30 for the night.
Each paid $10 for the room.
Later that evening the Manager said that the clerk overcharged the persons.
It should have been $25 for the night.
The clerk was given $5 to return to the 3 guys.
He figured that 3 into 5 wouldn't go.
So he gave each person 1$ in refund.
Each person gave $10 for the night.
Each person got back $1.
Do the math!
Each person gave $10.
Each person got back $1.
The clerk kept $2.
3x9=27.
The $2 the clerk kept= 29.
Whatever happened to the other dollar?
Posted by John Q. Public on August 13, 2007 at 9:59 am
I could answer but probably shouldn't.

I should also have placed a "spoiler warning" on my first post.
Posted by John Q. Public on August 13, 2007 at 10:02 am
Hmmmm... well, here is a hint for readers, without actually giving it away. The trick is to pay attention to where the money ends up, not who paid how much.
Posted by Jangaly_x on September 5, 2007 at 6:45 am
In the second arrangement the hypotenuse is not in the straight line. This results the second arrangement takes more area, to compensate must a squqre must be left.
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