The squares marked A and B are the same shade of gray

Wrote Kathryn Schulz in her very entertaining book, "Being Wrong: Adventures in The Margin of Error":

This is one of my favorite optical illusions, not because it is particularly dazzling but because it is particularly maddening.... If you think of this image as a checkerboard, then all the "white" squares that fall within the shadow of the cylinder (like B) are the same color as all the "black" squares that fall outside the shadow (like A).

You don't believe me, for the very good reason that you do believe your eyes, and your eyes are telling you that these squares look completely different. Actually, it's not your eyes that are telling you this; it's a handful of interpretive processes.... When it comes to determining the color of objects around us, our visual system can't afford to be too literal. If it were, it would do nothing but measure the wavelength of light reflecting off a given object. In that case, as the psychologist Steven Pinker has pointed out, we would think that a lump of coal sitting in bright sunlight was white, and that a lump of snow inside a dark house was black. Instead, we're able to correct for the presence of light and shadow so that the coal still appears fundamentally black and the snow still appears fundamentally white.

One way we do this is through local contrast. In nature, if something is lighter than its immediate surroundings, it's probably light in an absolute sense, rather than just because of the way the sun is or isn't striking it. That's one reason why, in this illusion, we read Square B (which is lighter than the dark checks around it) as light, period. The same phenomenon applies in reverse, so that we read Square A (which is darker than the squares around it) as dark, period. This interpretation is reinforced by several other interpretive processes, including the fact that we automatically adjust for cast shadows, mentally lightening whatever objects they fall on — in this case, Square B.

The net effect of these visual "corrections" is an illusion that is absolutely unshakeable. When I first saw it, I was so incredulous that I finally took a pair of scissors and cut the picture apart — whereupon, lo and behold, the A and B squares became indistinguishable from each other. In an effort to discourage you from mutilating this book, I offer a second image:

Not quite as convincing as slicing and dicing, perhaps, but a good start. (If you must cut it apart yourself to be persuaded, the original image... is available on the website of its creator, Edward Adelson, a professor of vision science at MIT).

Adelson's proof is here.

More?

Your wish is my demand: here he explains why the illusion works.