Comments on: The missing square: Where does it come from? http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/ School Safe Puzzles and Games for Kids of all ages Wed, 01 Feb 2012 04:15:27 +0000 hourly 1 http://wordpress.org/?v=3.0.2 By: fanfan http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-84416 fanfan Wed, 23 Feb 2011 21:24:34 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-84416 It is cunningly simple. The slopes of the blue and red triangles are different. The blue triangle has a steeper slope than the red. Hence in the upper diagram, the hypotenuse is slightly concave, ie., it is not a straight line. In the lower diagram, when the blue and red triangles are switched, the hypotenuse becomes convex, or bulged out. Therefore, the area of the bottom convex full triangle, including the missing square, is larger than the upper concave triangle. The difference in the areas, caused by the concavity and convexity of the hypotenuse, because of the difference in slopes of the blue and red triangles, IS THE EXTRA SPACE TAKEN BY THE LOWER DIAGRAM INTO WHICH THE MISSING SQUARE HAS DISAPPEARED. It is cunningly simple. The slopes of the blue and red triangles are different. The blue triangle has a steeper slope than the red. Hence in the upper diagram, the hypotenuse is slightly concave, ie., it is not a straight line. In the lower diagram, when the blue and red triangles are switched, the hypotenuse becomes convex, or bulged out. Therefore, the area of the bottom convex full triangle, including the missing square, is larger than the upper concave triangle. The difference in the areas, caused by the concavity and convexity of the hypotenuse, because of the difference in slopes of the blue and red triangles, IS THE EXTRA SPACE TAKEN BY THE LOWER DIAGRAM INTO WHICH THE MISSING SQUARE HAS DISAPPEARED.

]]> By: mcmike http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-83098 mcmike Thu, 10 Jun 2010 09:11:54 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-83098 This has to do with Curry's Paradox and the Fibonacci Sequence. The figure you are looking at is not actually a triangle! If you look at the two smaller triangles, one has a slope of 3/8 and one 2/5. If they have different slopes, then the hypotenuse is not actually a straight line segment, but rather two line segments. The area shown above is actually the same exact area you are seeing when the pieces are mixed up. Every missing square triangle like this has two triangles with slopes that are just ratios taken from the Fibonacci Sequence 1,1,2,3,5,8,13. Notice that 2/5 and 3/8 are both ratios of a(sub n)/a(sub n +2). Its quite neat! This has to do with Curry’s Paradox and the Fibonacci Sequence. The figure you are looking at is not actually a triangle! If you look at the two smaller triangles, one has a slope of 3/8 and one 2/5. If they have different slopes, then the hypotenuse is not actually a straight line segment, but rather two line segments. The area shown above is actually the same exact area you are seeing when the pieces are mixed up. Every missing square triangle like this has two triangles with slopes that are just ratios taken from the Fibonacci Sequence 1,1,2,3,5,8,13. Notice that 2/5 and 3/8 are both ratios of a(sub n)/a(sub n +2). Its quite neat!

]]> By: toxicasset http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-81989 toxicasset Mon, 21 Dec 2009 17:17:32 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-81989 'Hypo' this 'hypo' that.....The area of the two triangles are EXACTLY the same you foolish people.... aaaaarrrggghhhh ‘Hypo’ this ‘hypo’ that…..The area of the two triangles are EXACTLY the same you foolish people….


aaaaarrrggghhhh

]]> By: toxicasset http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-81988 toxicasset Mon, 21 Dec 2009 17:11:17 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-81988 This is an easy square area problem.... It's the resulting rectangular area you need to focus upon! They are different to one another. One is a 5 x 3 and the other is an 8 x 2. One has a square area of 15 and the other a square area of 16.... Simplezzzzzz This is an easy square area problem….


It’s the resulting rectangular area you need to focus upon!
They are different to one another.
One is a 5 x 3 and the other is an 8 x 2. One has a square area of 15 and the other a square area of 16….


Simplezzzzzz

]]> By: bilbao http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-79767 bilbao Mon, 18 May 2009 10:44:44 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-79767 Original source: According to Martin Gardner, a New York city amateur magician Paul Curry invented this popular paradox in 1953 Original source:
According to Martin Gardner, a New York city amateur magician Paul Curry invented this popular paradox in 1953

]]> By: Tommy http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-59828 Tommy Tue, 27 Nov 2007 21:52:20 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-59828 I've been pretty good at math for almost as long I remember. My dad used to teach math in China. Now, he's a bio statistician. I’ve been pretty good at math for almost as long I remember. My dad used to teach math in China. Now, he’s a bio statistician.

]]> By: RK http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-59603 RK Tue, 27 Nov 2007 02:57:08 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-59603 Tommy- just curious, when did family/teachers realize you were so good at Math? Anyone in your family have a math/engineering educational background? Tommy- just curious, when did family/teachers realize you were so good at Math? Anyone in your family have a math/engineering educational background?

]]> By: Tommy http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-59563 Tommy Mon, 26 Nov 2007 21:59:10 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-59563 I understood the Monty Hall problem when I was 8 or 9. This may be caused by the same thing as the Flynn Effect, or this may because I rock at math. I actually do it competitively. I ranked 33rd in the nation in MATHCOUNTS last year, though I should of done a lot better. I understood the Monty Hall problem when I was 8 or 9. This may be caused by the same thing as the Flynn Effect, or this may because I rock at math. I actually do it competitively. I ranked 33rd in the nation in MATHCOUNTS last year, though I should of done a lot better.

]]> By: Ashleigh http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-30114 Ashleigh Sun, 26 Aug 2007 04:01:19 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-30114 I have only taken 1 math class per day but am the smartest in my math group or class I have only taken 1 math class per day but am the smartest in my math group or class

]]> By: RK http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-28417 RK Tue, 14 Aug 2007 00:55:52 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-28417 Ashleigh- what math classes have you taken at 10?!? Ashleigh- what math classes have you taken at 10?!?

]]> By: Ashleigh http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-28416 Ashleigh Tue, 14 Aug 2007 00:51:27 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-28416 That was easy and i am only 10!! That was easy and i am only 10!!

]]> By: Abhijit http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-21351 Abhijit Fri, 29 Jun 2007 00:59:48 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-21351 The space allotted to the portions colored yellow and green taken together increases by by one small square . This is due to differential ( small ) increase in the total area of the triangle after rearrangement of the different sectors i.e, red , blue green and yellow . However only minute obseravation lets us see the change in area covered by the different sectors . The space allotted to the portions colored yellow and green taken together increases by by one small square . This is due to differential ( small ) increase in the total area of the triangle after rearrangement of the different sectors i.e, red , blue green and yellow . However only minute obseravation lets us see the change in area covered by the different sectors .

]]> By: sophie http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-7590 sophie Sat, 05 May 2007 02:59:06 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-7590 the top shape is quadrilateral not triangular the top shape is quadrilateral not triangular

]]> By: Jean Paul http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-2292 Jean Paul Thu, 15 Feb 2007 09:40:45 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-2292 The different slopes is the key factor. The area of a 13x5 triangle is 32.5 but the sum of the area of the different pieces is only 32. Thus rearranging the pieces one can either compensate the missing .5 making in look compact (1st picture) or build a hole (2nd picture). The different slopes is the key factor. The area of a 13×5 triangle is 32.5 but the sum of the area of the different pieces is only 32. Thus rearranging the pieces one can either compensate the missing .5 making in look compact (1st picture) or build a hole (2nd picture).

]]> By: Lord Gordo http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/comment-page-1/#comment-1633 Lord Gordo Tue, 23 Jan 2007 02:53:40 +0000 http://www.smart-kit.com/s283/the-missing-square-where-does-it-come-from/#comment-1633 The blue triangle is alot shorter than the big red triangle, therefor for the blue to line up equal with the red is if the yellow rectangle is moved back and leaves a empty box after the yellow rectangle. The blue triangle is alot shorter than the big red triangle, therefor for the blue to line up equal with the red is if the yellow rectangle is moved back and leaves a empty box after the yellow rectangle.

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