Can a neural network work out the concept of distance? Imagine a game where it is a black screen apart from a red pixel and a blue pixel. Given this game to a human, they will first see that pressing the arrow keys will move the red pixel. The next thing they will try is to move the red pixel onto the blue pixel.

Give this game to an AI, it will randomly move the red pixel until a million tries later it accidentally moves onto the blue pixel to get a reward. If the AI had some concept of distance between the red and blue pixel, it might try to minimize this distance.

Without actually programming in the concept of distance, if we take the pixels of the game can we calculate a number(s), such as "entropy", that would be lower when pixels are far apart than when close together? It should work with other configurations of pixels. Such as a game with three pixels where one is good and one is bad. Just to give the neural network more of a sense of how the screen looks? Then give the NN a goal, such as "try to minimize the entropy of the board as well as try to get rewards".

Is there anything akin to this in current research?
Answer

I'm going to take your question at face value, and go really deep into this topic. 

Yes, they can. The typical human mind can. But consider the human mind. Millions, if not billions, of neurons. In fact, one can consider distance as a human concept, simply a theory developed from interactions with the world. 

Therefore, given a year or two, with a ton of neurons on your hand, you could replicate this scenario. That is if your computer is as parallel as the human mind. The short explanation is that the human mind is very parallel.

However, it would be simpler to calculate the distance with a program, not an AI, and simply feed the result to the AI that would make the decisions.

Consider the amount of time you have spent looking at a screen. If you can tell the (approximate) distance between two pixels, so can a Neural Network, as you are one. However, add the amount of time you have spent alive and learning into the equation, and it becomes a disaster. 

Further reading

The human brain is parallel

This is a result of the fact that all of the neurons in the human brain are independent of each other. They can run true simultaneous actions, thus making the action of interpreting images and such much easier, as blocks of neurons can "think" independent of the operations of the others, limiting what would be "lag" to a minuscule amount.