Map Coloring: Questions 11–15

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Welcome. We have six countries — R, S, W, X, Y, and Z — arranged on a map with specific shared borders. The one fundamental rule is simple: any two countries that share a border must be colored differently. Keep that rule in mind as we work through every question. Look closely at W, X, and Y. W borders X, W borders Y, and X also borders Y directly. These three countries form a triangle, meaning each pair is adjacent, so all three must be different colors. This triangle alone proves we need at least three colors for the entire map. Now let us answer Question 11. Which two countries must always be different colors? S and Z share a direct border on our map, so by the fundamental rule they can never have the same color. The answer to Question 11 is C: S and Z must be different colors. Question 12 asks: if X and Z are the same color, which other two countries must also share a color? Y borders X so Y cannot use X's color. R borders Z so R cannot use Z's color. Both R and Y are also adjacent to W, leaving only one remaining color for each of them, forcing R and Y to be identical. Question 12 answer is A. Question 13 asks whether R and S can share the same color. Notice there is absolutely no border drawn between R and S on our map — they are separated by other regions. Because they are not adjacent, the coloring rule does not apply between them, and they are perfectly free to share a color. Question 13 answer is A. Question 14: can Z share the same color as W? W is surrounded by R, S, X, and Y — four neighbors — but W and Z share no border at all. Since they are not adjacent, the rule places no restriction between them, and Z is free to take whatever color W uses. Question 14 answer is E. Finally, Question 15 asks for the minimum number of colors needed. The W–X–Y triangle demands three colors. A valid coloring of all six countries can always be completed with exactly three colors, and in different valid arrangements W, X, or Y each can be the sole country using one particular color. The minimum is three, and Question 15 answer is D. To recap: Q11 C, Q12 A, Q13 A, Q14 E, Q15 D. Always identify every border, look for triangles, and distinguish what must be true from what can be true.