# How come $32.5 = 31.5$?

Below is a visual proof (!) that $32.5 = 31.5$. How could that be?

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@Peter: why not? This is not mathoverflow, and basic to one person is not basic to another. – Simon Nickerson Jul 21 '10 at 7:58
If this site is to become the best and most complete resource on Mathematics for non-mathematicians, I'm guessing at some point someone will be looking for an answer to this question here, which makes this a valid question. – Edan Maor Jul 21 '10 at 8:11
+1 and voted to reopen. In my opinion, the motives of the OP are a non-issue compared to the goal of building a math knowledge base. Like it or not, this question/puzzle/joke has become an internet staple and there are sure to be people interested in an explanation. – e.James Jul 21 '10 at 8:34
@Harry please stop making nonsense edits -- no matter if this question should be closed or not, the tag [tag-removed] is totally useless – balpha Jul 21 '10 at 9:42
@Harry: "This question contains no mathematical content."? Um, okay, how do you explain the apparent paradox then? Even if you don't like the question (I'm still not sure why), it's definitely a math question...geometry in particular. – Beska Jul 21 '10 at 13:05

It's an optical illusion - neither the first nor second set of blocks actually describes a triangle. The diagonal edge of the first is slightly concave and that of the second is slightly convex.

To see clearly, look at the gradients of the hypotenuses of the red and blue triangles - they're not 'similar'.

gradient of blue triangle hypotenuse = 2/5
gradient of red triangle hypotenuse = 3/8

Since these gradients are different, combining them in the ways shown in the diagram does not produce an overall straight (diagonal) line.

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The word you call "gradient" is more familiar to me as "slope", but I guess it's just regional variation. – ShreevatsaR Aug 5 '10 at 16:25
It's not exactly a coincidence that these are ratios of Fibonacci numbers. That's a good way to end up with ratios that are quite close to each other. In this case, both are approximants to $\frac{3-\sqrt{5}}{2}$, the limit of $\frac{F_n}{F_{n+2}}$. – alex.jordan Feb 1 '15 at 8:42

Since this question has been bumped up, and some people find pictures helpful, may as well supply illustrations of the explanation.

The area of the red triangle is $\frac12(8)(3) = 12$ and that of the blue triangle is $\frac12(5)(2) = 5$. The area of the yellow and green regions are clearly 7 and 8 respectively, so the total coloured area is 12 + 5 + 7 + 8 = 32 which is less than 32.5, the area of a 13×5 triangle. Indeed, in the figure below you can see that the hypotenuses of the red and blue triangles "dip" below the actual hypotenuse of the large triangle; this accounts for the 0.5 difference in area.

Similarly, in the second figure, the hypotenuse of the red and blue triangles are above the actual hypotenuse, which accounts for the 0.5 difference in area again.

The difference between the two figures is a thin parallelogram with vertices at the endpoints of the hypotenuses in both figures (coloured pink below), which has area exactly 1. (If it looks much less, that's the nature of the optical illusion.)

Also, there are some links to similar figures at Wikipedia's article on this missing square puzzle.

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@pushpen.paul: Please don't edit answers unnecessarily, and please don't allege "problems" simply because you may have different preferences. There is no requirement on this site to use math markup everywhere (it's enough to be readable; I have already used math mode in the answer where it was absolutely necessarily), and certainly no requirement to use American spellings. – ShreevatsaR Jul 20 '14 at 17:37

Overlay the two triangles to see the difference. This is known as the missing square puzzle.

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So, i think it's a known puzzle. – ABC Apr 22 '13 at 14:46
@exploringnet: A very well known, indeed! (+1) for the great illustration. – Inceptio Apr 22 '13 at 14:47
@exploringnet oh, indeed it is. – oldrinb Apr 22 '13 at 14:47
What software did you use to create that gif? – Math Gems Apr 22 '13 at 22:55
@MathGems unfortunately it is not mine. – oldrinb May 7 '14 at 22:17

Surprisingly, the general case remains to be explained.

Consider the recursion $F(n+2) = F(n+1) + F(n)$, with $F(1) = F(2) = 1$
producing the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ...

Let the blue triangle have horizontal and vertical sides of length $F(n+1)$ and $F(n-1)$, respectively.
Let the red triangle have horizontal and vertical sides of length $F(n+2)$ and $F(n)$, respectively.
So that
in the upper figure the two triangles create a rectangle with an area of $F(n)F(n+1)$
and
in the lower figure the two triangles create a rectangle with an area of $F(n-1)F(n+2)$

By the Fibonacci identity $F(n-1)F(n+2) - F(n)F(n+1) = \pm1$
we can see that the two rectangles are bound to differ in area by exactly one unit.

As n increases the gradient/slope of each triangle tends to the square of the Golden Ratio $\phi$,
one from above and one from below.
Thus, as $\phi^2$ = 0.3819660112...
2/5 = 0.4 > $\phi^2$
3/8 = 0.375 < $\phi^2$
5/13 = 0.3846... > $\phi^2$
etc.

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As an aside, it's relatively harder to dissect the F(n)×F(n+1) and F(n-1)×F(n+2) rectangles into two shapes (which leave a "hole" in the second case), even though they differ in area by 1. (At least, I couldn't see something simple.) – ShreevatsaR Jun 4 '11 at 4:35

No, this is not a proof of your statement. If you look very closely, you will see that the hypotenuse of the triangles aren't straight. You can also verify this algebraically by calculating the internal angles of the triangle using trigonometry.

The other way to demonstrate the non-straightness of the hypotenuse is in the fact that the red and blue triangles aren't similar, which can be seen by the difference in the ratios of their width and height. For the blue triangle, this is 2/5 and for the red triangle this is 3/8. As they are both right angled triangles, these ratios would be the same for similar triangles and the fact that they aren't means the interior angles are different.

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+1 for the words "look closely"... well, how close should one be?? I was kidding :) – awllower Apr 5 '11 at 10:13

The red and blue triangles are not similar (the ratios of the sides are 3/8 = 0.375 and 2/5 = 0.4 respectively), so the "hypotenuse" of your big triangle is not a straight line.

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