Imagine an island composed of several connected countries with no enclaves or exclaves. We are concerned with coloring a map of the island with as few colors as possible.
If two neighboring countries merge their border dissolves and they become the same color.
Begin with an island that can be two colored.
One or more pairs of countries merge and now the new map of the island can only be three colored.
Again, one or more countries merge and now the map of the island requires four colors.
Find the smallest number of starting countries for which this is possible as well how they are joined.
A diagram like this example of a 3 color map should be easy enough to read:
My picture has 7 initial countries, but it is not so easy for me to draw.
Start with an island shaped as equilateral triangle. Connect the midpoints of the 3 sides, forming 4 countries. Similarly, divide the middle country into 4, by joining the midpoints of its borders. Now you have 7 countries, which only require two colors.
Merge one of the larger outer countries with a bordering country. Now three colors are required.
Merge each of the other two larger countries with a bordering original country, and now 4 are required.
Of course, there are lots of other topologically equivalent shapes, but none (I think) that are like the grid that the puzzle was suggesting.
Were you trying to fool us, Jer?