Poggle is similar to a hand-held electronic game that was produced a decade or so ago. Press a button and lights change for several buttons. The player continues until all buttons are lit in particular manner. I believe it meant going from one colour to another.
In this version of Poggle the objective is to get all squares from 'off' (black) to 'on' (white). Selecting any square will change its colour in a set sequence, as well as the colour of its neighboring square above, below, left and right. As the playing field does not wrap to the opposite sides, squares on the border are treated differently.
The variants that this program will handle are changes in the size of the field (options ranging from 3-by-3 to 7-by-7) and number of colours that the squares will cycle through. The maximum numbers of colours mean that squares will go from black - red - green - blue - white and to black again.
Like I wrote earlier, there must be a way of proving that a solution exists. I just have no clue as to where to start looking.