How can I implement an algorithm for a matrix where each cell can have one of two values, with hints

Question

Firstly, sorry for the vagueness of the title.

I don't know if there is a name for this sort of algorithm.

I am interested in finding out how this scenario could be implemented.

Say I have a matrix (columns and rows) where each "cell" can have one of two values ("on" and "off").

Each cell's "background" should also display a number that tells how many of the adjacent cells are "on".

For example, if every cell should be "on":

A	B	C		A	B	C
4	6	4	-->	on	on	on
6	9	6		on	on	on
4	6	4		on	on	on

This is pretty simple to implement. The question is: what's the minimal amount of tips I can have, and how can I implement this?

In this same example, it would be enough to have:

A	B	C		A	B	C
			-->	on	on	on
	9			on	on	on
				on	on	on

It can get trickier:

A	B	C		A	B	C
			-->	on	?	off
4		1		on	?	off
				on	?	off

In this last example, the matrix would be larger and have other hints that later on would help determine the remaining cells.

So how can this be achieved?