Many of my readers are intelligent and scientifically-literate people. So today I’d like to share this cute paradox I came up with during my vacation last month. I was discussing some computer science topics with my buddy Luqman when I got this idea.

The paradox itself not that hard to resolve; one can argue that it’s not an actual paradox but rather a layman’s misunderstanding of the nature of “information” (I’m not surprised if someone has thought of this before). The paradox is very simple, and this is how it works: a seemingly reasonable hypothesis results in an absurd conclusion, so there must be a logical error somewhere. Your job is to spot where the logical error occurs.

If you can resolve the paradox, please provide your explanations in the comments field below. I put this up on /sci/ and it took less than a minute for a forum member to solve it.

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Consider all the photos you can take using a standard digital camera. Each photo is determined uniquely by the color of each pixel (other properties like brightness, contrast, etc are simply a product of pixel coloring and so are not relevant here).

Since the photo is digital, there is a finite number of pixels, say M (roughly in the order of 1 million), and each pixel can be colored in a finite number of colors, say N (assuming 8-bit color, we have 256 x 256 x 256 possible colors). We do not care about the actual values of M and N. The only important thing to keep in mind is that while M and N are huge numbers, they are finite.

By a basic counting argument, the number of all possible images is N^M (N raised to the power of M). A huge number, but still finite. We call this number K.

Now, all possible images that can be taken with the camera belong to this collection of K photos. Loosely speaking, this collection of K photos contains *all possible images in the universe*, at the resolution which the camera allows.

Suppose I’d like to perform this process: First, I write the number 1 on a paper and take a photo of that paper. Then I write the number 2 on a paper, and take a photo of that paper. Then I repeat this process, going on and on until I reach the number K+1 (it might take a very long, but still finite, time). Now I have generated K+1 photos. This is a contradiction to our earlier statement that there are only K possible photos that can be recorded with the camera.

Where does my argument break down?

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Pic related:

The dimensional resolution and the colour resolution is simply insufficient to accommodate >K combinations of numbers.

We can use a trivially easier case to illustrate – think of a one-pixel monochrome camera. Try to use that camera to take pics of the numbers 1, 2 and 3. 🙂

mind blowing fact of the day: a 17×17 pixel monochrome camera can generate more images (2^289) than the number of atoms in the entire observable universe (~10^80).

Good writhing.. and if any is attending to buy a modern digital photographic camera with effective device characteristic, here is a short review about the Casio Exilim 10mp camera.

http://www.squidoo.com/casioexilim10mp

I sure do writhe well.

not so sure about this but wouldn’t the number of all possible images be 2^((MxN)-1)) instead?

number of possible images = number of possible colors ^ number of pixels.

if we have 2 colors and 5 pixels, we can choose to color each pixel in 2 ways, and the choices are independent, so the total number of possible images is 2 x 2 x 2 x 2 x 2 = 2^5 = 32.

bwah. yeah somehow instead of 2x2x2x2x2 i thought it’s concatenating 2-bits five times. brainfart.

ill give it a try…

i think it breaks down in the last process. for the given number of pixels M and possible number of colors N, you can only write the numbers and take its picture up till a MAXIMUM of K pictures. That is, for the given picture resolution, you won’t be able to write the number K+1 and snap its picture, no matter how hard you try. This is because the number would be so large, that the resolution of the picture won’t be able to accommodate for the number of pixels that it has to capture. It’s like trying to describe the number 10^10 in 8 bits; there isn’t enough space.

is my answer even close?

Azizi is right (and so is Chang Yang).

It is not difficult at all; in fact the only “difficulty” is when you denote a super large number by a letter K, it feels like a small number psychologically (unless you have done a lot of mathematics, and feel comfortable dealing with large numbers). Psychologically, when we use K to denote a number, one “feels” that writing 1 to K+1 is a process that takes a “limited” amount of time, something that can be done at least theoretically (in practice, it would take longer than the time from big bang to heat death of the universe, even if we are able to write each number in, say, one nanosecond).

the whole point is that this process is not only physically impossible (we are dealing with thought process after all), it is mathematically impossible as well.

K is a superhuge number that would take more than M digits — the nearest integer to log_10 K + 1 to be exact — and that is if K is written in base 10. The best possible way is if we write our numbers using base N (using each pixel as one digit-place and use colors as “digits”) and even then the best we can do is K. (It would be interesting to think that using this “number base”, you can represent a unique number from 1 to K by a photo of a cat).

The mathematical solution is simple, but not as psychologically satisfying: by the pigeonhole principle, if we write down numbers 1 to K+1, then two of them will have similar images and therefore indistinguishable.