[SpaceAce]

This is actually a two-parter. The obvious answer to part one is that all numbers are equally likely to show up but I don't think it can be quite that simple because of the "3"s (see description).

Part 1:
Take an X by Y grid. Let's put X and Y at 15 and 15. You have tiles numbered 1-9 to fill the board with. You pick the tiles blind and every tile has an equal chance of coming up. You fill the 15x15 grid with tiles in this manner. You then add the value of the tiles following these rules:

- The individual tiles mean nothing. Only groups of tiles added together count for anything in this scenario.
- The tiles are taken at face value. A tile labeled "9" is worth nine and so on.
- You only add the values of non-diagonal neighbors. So, the maximum number of neighbors a tile can have is four (top, bottom, left, right) and tiles that are along any edge of the board have only three neighbors. The four corner tiles have only two neighbors each. In this manner, the minimum number a group of tiles can add up to is three (a corner tile worth one with two neighbors also worth one each). The most a group of tiles can add up to is 45 (a nine with four neighbors of nine each).

My question is, if you randomly fill this board with the tiles numbered 1-9, is there any number that is the most likely to show up in the groups? It seems that I should expect to see less threes than anything else because there are only four possible places to make a three, but what about the other numbers?

Part 2:
If the numbers will not be evenly distributed, how do you calculate the likelyhood of a given number, or the number of times you can expect to see it in any given grid?

Thanks,
SpaceAce