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Printjmc
counting and probability senior
Problem
Regular octagon has its center at . Each of the vertices and the center are to be associated with one of the digits through , with each digit used once, in such a way that the sums of the numbers on the lines , , , and are all equal. In how many ways can this be done? 
Solution
Let denote the common sum of the numbers on each line. Then gives the sum of all the numbers but with counted four times. Since the sum of the numbers on the octagon must be we have (where represents the number written at that vertex). Thus, must be a multiple of , which occurs exactly when
If then so It follows that the sum of each pair of diametrically opposite vertices is so we must pair up the numbers , , , and There are ways to assign the four pairs, and then ways to assign the two numbers in each individual pair. Therefore, in the case , there are ways to label the vertices.
The cases and are the same, and also produce valid ways. Thus, the total number of ways to label the vertices is
If then so It follows that the sum of each pair of diametrically opposite vertices is so we must pair up the numbers , , , and There are ways to assign the four pairs, and then ways to assign the two numbers in each individual pair. Therefore, in the case , there are ways to label the vertices.
The cases and are the same, and also produce valid ways. Thus, the total number of ways to label the vertices is
Final answer
1152