Consider an equilateral triangle with points located at each vertex and at each midpoint of a side. (See picture.) This problem uses the set of numbers {1, 2, 3, 4, 5, 6}. Place one number at each point. Call the sum of the three numbers along any one side (two vertices and one midpoint) a “Side Sum.” There are arrangements of the numbers so that the sum of the numbers along any side is equal to the sum of the numbers along each of the two other sides. We will call this arrangement an Equal Side Sum solution.
1) Show that more than one Equal Side Sum solution exists.
a. For which numbers are Equal Side Sum solutions possible? (Show by giving examples for each Equal Side Sum solution that is possible.) Comment on how you obtained these solutions.
b. What is the smallest number for which there is an Equal Side Sum solution? Why?
c. What is the largest number for which there is an Equal Side Sum solution? Why?
2) Is there more than one Equal Side Sum solution for the same number? (To answer this question, you will need to be precise as to what you mean when you say that two Equal Side Sum solutions are the same or are different.) Explain your answer.
Extra Credit: It is possible to generalize The Triangle Game to create a similar game involving other polygons. Describe such a game. Are you able to find any solutions to your new game? Worth up to two extra credit points
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