Popular Missing Square Puzzle

How many number of times you have come across this image?

By now, you might have accepted the solution.

"The hypotenuse is bulged in bottom figure which accounts area equal to missing square!"

But is this really a solution to this puzzle? The issue illustrated in clear manner here!

And my opinion is here!

And how creator did manage to create that 'missing' square is illustrated here!

 

Visual Illusions Psychology

Which monster looking bigger? The one which is chasing the other? Right?

Bigger Vs Smaller

Now look at this. What's your take now?

Equal Heights

Both are of same size. A 3D tunnel built around deceives your visual system & totally changes how it perceive things.

Now take a look at this. Are these straight lines or not? Definitely not! Right?

Straight or Curved

But you are totally wrong! Have a look at this.

Only Straight Lines

Again white & black checks surrounded in certain disorder changes how you perceive things.  

Checker Shadow illusion

Which square is darker here; A or B?

A is darker or B

The A appears darker than B here. But that's not the reality. Both shares equal levels of brightness & gray levels. A printer would you same ink to print those squares. 

Equally Darker

What's challenge to visual system here?

The task given to your visual system here is to determine the color of the object & here it is gray level. But shadow of cast plays here big role. The square under shadow reflecting less light than the one is not in shadow. Visual system need to do several tricks to compensate that shadow effect & determine the exact color of the square under shadow.

What tricks visual system uses?

1. Local contrast : Irrespective of shadow, a lighter check surrounded by more lighter checks is perceived as a check with more average lightness & vice versa. In the figure, the light check in shadow is surrounded by darker checks. Thus, even though the check is physically dark, it is light when compared to its neighbors. The dark checks outside the shadow, conversely, are surrounded by lighter checks, so they look dark by comparison.

2. Shadows often have soft edges while paints (here checks) have sharp edges. The visual system tends to ignore gradual changes in light level, so that it can determine the color of the surfaces without being misled by shadows. In this figure, the shadow looks like a shadow, both because it is fuzzy and because the shadow casting object is visible.

3. The "paintness" of the checks is aided by the form of the "X-junctions" formed by 4 abutting checks. This type of junction is usually a signal that all the edges should be interpreted as changes in surface color rather than in terms of shadows or lighting.
 
Conclusion :

The effect illustrates success rather than failure of our visual system. The illusion asking our visual system to act as a light meter; but it's definitely not built for it. What's the success of our visual system here is it breaks the information into meaningful components & then perceive different aspects of objects.



Courtesy : Wikipedia
 

How Missing Square Puzzle Created?

That's how it was missed! 

Slope of diagonal must be 5/13 = 0.385

If we keep heights of red & green triangles constant as 3 & 2 respectively

then their bases should have been.

5/13 = 3/Base of red triangle 

Base of red triangle = 7.8

And,
 

Base of green triangle = 13 - 7.8 = 5.2

Creator of this optical illusion cleverly restricted 5.2 to 5 & extended 7.8 to 8.

Or we can say he altered heights of those triangles by keeping bases 8 & 5.

Now creator shuffles those colored blocks like shown below.

Missing square puzzle

Pink & Blue rectangles shuffled diagonally. Purple rectangle of size 15 

colored blocks is replaced by orange rectangle of size 16. Obviously orange

rectangle would have 1 white block & 15 colored blocks!

 

Origin of Missing Square Puzzle

First view this! 

Actually the diagonal drawn here isn't a true diagonal.....It's slope must be 5/13 = 0.385.

Two hypotenuses (of two triangles) have different slopes; red one having 3/8 = 0.375 &
green one has 2/5 = 0.4.

So both together can't form true diagonal of this rectangle neither hypotenuse of colored triangle.

So we need to calculate area of each colored shape differently & then add to get area of colored region.

Area of colored region = 12 + 5 + 15 = 32..... Not 32.5

So if it was halved by perfect diagonal then 32.5 - 32 = 0.5 of area would have been painted more.

This is half of that white space.This is also reason why non painted part of top rectangle covers area 12 + 5 + 16 = 33 i.e. 0.5 than expected 32.5 because of imperfect diagonal.

And if we calculate area of other part (non painted) of bottom rectangle as 12 +5 +15  = 32 ; again less by 0.5 if there was perfect diagonal.

Since it is not a perfect diagonal that white space is 'missing' 0.5 of colored area & 0.5 of white space is 'taken' from other part of rectangle.

That's why 1 white space there!