The Mathematics Education Department
EMT 725, Lisa F. Garey

Coloring Circles

Click here for the Problem Statement

This problem can be analyzed as cases.

Case I: Using 2 colors. Choose a color for the 1st circle at the top left corner. Then choose a 2nd color for the 3 circles connected to the 1st one. Let the last circle be the same color as the 1st one.

After choosing the 1st color, there are 2 choices for the three circles. The last circle is determined by the 1st one. Since there are 3 colors to choose from, there is a total of 6 patterns for case I.

Case II: Choose a color for the 1st circle at the top left corner. Then choose a 2nd color for the 3 circles connected to the 1st one. Unlike in case I, choose a different color for the last circle than the 1st one.

As in case I, after choosing the 1st color, there are 2 choices for the three circles. The last circle is determined by the 1st one. Since there are 3 colors to choose from, there is a total of 6 patterns for case II.

Case III: Choose a color for the 1st circle at the top left corner. Then choose a 2nd color for the circle connected diagonally to the 1st one. Use yet a different color for the 2 circles connected orthogonally to the 1st one. This means there is only one choice since no two connected circles can have the same color. Then there are 2 choices for the last circle.

There are 3 choices for the 1st circle. There are 2 patterns for the next three circles. There are 2 possibilities for the last circle. Thus, there are 3(2)(2)=12 patterns for case III.

Case IV: Choose a color for the 1st circle at the top left corner. Then choose a 2nd color for the circle connected diagonally to the 1st one. Use 2 different colors for the 2 circles connected orthogonally to the 1st one. Then the last circle has to be the same color as the 1st one.

There are 3 choices for the 1st circle. There are 2 choices for the next one with 2 possible patterns for the other two circles. The last circle is determined by the first one. Thus, there are 3(2)(2)=12 patterns for case IV.

We have exhausted all the possibilities. Thus, there are 6+6+12+12 = 36 patterns altogether.

Comments: The more colors available to choose from, the more patterns there are. The rules also play a part in determining the # patterns allowed. For instance, the # total possibilities of patterns would increase dramatically if rule 3 (no circles connected by a single line may be of the same color) is eliminated. The configuration of the circles also determine the patterns.

Extensions:

• Calculation of the probability of various events can be calculated e.g. what is the probabilty that the 3 circles along the diagonal will have the same color? What about the other diagonal?
• Investigation of the sample spaces in other situations mentioned above.

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