This is easier to draw than to explain, but I'll try. Imagine N-countries shaped generally as concentric rings. The outer-most ring, however, has an isthmus into the very center, forming both "ring" 0 and "ring" N with ring 0 being the center, and N being the outer most. Let's state this isthmus occurs at the bottom of the ring for convenience. Now, if ring 1 has a parallel isthmus that runs along the left of N's and parallel to that of of N's until it reaches N, and ring 2 has its own that runs in the same manner until it hits ring N-2, etc. Then, consider if on the right side, you do the opposite: N-1 extends until it hits ring 1, N-2 until it hits ring 2.
It's a rather easy formula to produce a "map" that requires N colors if no two colors are to touch.