Type a counter-example that would have to exist in order for the conclusion to be false.
Every map that has ever been drawn can be colored with four colors so that no two regions colored alike touch at more than one point.
Conclusion: Every map can be colored with four colors.
counter- example:
To find a counter-example for the given conclusion, we need to find a map that cannot be colored with four colors as described.
A possible counter-example could be a map representing a scenario where there are five regions (such as countries or states) arranged in such a way that each region touches all the other regions. In this case, it would not be possible to color the map with just four colors, as there would be at least two regions that touch at more than one point.
To further explain how to find such a counter-example, one could think of a specific arrangement of regions or draw a sketch of the map to visualize the problem. By examining the arrangement of regions and their connections, it becomes clear that there is no way to color the map using only four colors without violating the given condition.
This counter-example shows that not every map can be colored with four colors, thus disproving the conclusion that every map can be colored with four colors.