Mathematical sequence

[krishkr@rediffmail.com]

Imagine there is a very very large room in the shape of a hollow cube. There are magic balls hanging in the air at fixed discrete positions of the room. No magic ball has another one exactly above it. If we take an imaginary horizontal plane of infinite area and pass through the cube, how can we be sure that the plane doesn't cut through any of the magic balls ?

The height of a magic ball is given as a function of its position. The distribution is such a way that some balls are at the same height while other are at different heights. The positions (x and y) and also the height (z) are discrete values (for simplicity, we can consider them positive integers).

Let the function be
z = axy + bx + cy
where a,b,c are positive integer constants. The positions (x-axis and y-axis values) and also the height (z) are discrete values (for simplicity, we can consider them positive integers).

If the ball distribution function was z=10xy+8x+4y, then it is impossible to have a z value of 15 or 21. So a plane at z=15 or z=21 would not cut any of the balls! In fact, in this case, any plane with a height (z = any odd number) would not cut through the balls. It is noticeable that there a some planes with height as even numbers that donot cut through the balls.

We do not want to find the heights of all the magic balls and compare it with the height of the horizontal plane, as that would be like trying all the possible combinations and would take very long time even on a computer.

Our aim is to find a fast method by which we can tell whether a given value of z (height) can be produced by any pair of (x,y) (positions). If a given z cannot be produced, then a plane at that height doesn't cut through any balls!
The question is also similar to finding whether a given number is present in a sequence produced by a function of two variables.

It would a great help if U could give me any suggestions to solve this problem. Thank You. (I have already tried evolutionary computing like GA,PSO,DE,SA etc. The method needs to be deterministic).

[DarkerLine]

Unless there is a pattern, there is obviously no way to do this short of checking the height of every ball. If there is a pattern, then it depends on the pattern.