Problem. Which letters can be drawn without lifting the pencil or double tracing? BCDLMNOPQRSUVWZ, not AEFGHIJKLTXY. Yeah this is just Eulerian tours, okay.

The letters that can be drawn without lifting the pencil or double tracing are B, C, D, L, M, N, O, P, Q, R, S, U, V, W, and Z. These letters can be traced along an Eulerian tour, which is a path that visits each edge (or line segment in this case) exactly once. The letters A, E, F, G, H, I, J, K, L, T, X, and Y cannot be drawn without lifting the pencil or double tracing because they require retracing some segments or lifting the pencil to complete the letter.

Problem. Four color theorem on CA map.
Three colors is not sufficient if there is an odd cycle whose vertices are all connected to some other vertex.

The Four Color Theorem states that any map on a plane can be colored using at most four colors in such a way that no two adjacent regions (countries) have the same color. In other words, four colors are always sufficient to color any map.

However, it is important to note that there are certain conditions where three colors are not sufficient. Specifically, if there exists an odd cycle in the map where all the vertices of the cycle are connected to some other vertex, then three colors alone cannot be used to color the map without adjacent regions sharing the same color.

This condition is known as an odd cycle with a “star” configuration. In this case, additional colors are needed to properly color the map without any adjacent regions having the same color. Therefore, to ensure a proper coloring in such cases, at least four colors are required.

Overall, the Four Color Theorem guarantees that for any general map on a plane, four colors are always sufficient to achieve a proper coloring, except for certain specific configurations such as odd cycles with a star configuration where additional colors may be needed.

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