Problem #9

The integers from 1 to n are arbitrarily assigned to the vertices of an n-gon. If two adjacent vertices are labeled i and j, the edge connecting them is labeled |i − j|. Let S denote the sum of the edge labels. An example with n = 6 is shown below yielding a value of S = 14.

For a given n

• What is the smallest possible value for S?

• How many different assignments of vertex values give this minimum? (Note: Assignments that differ by a rotation or a reflection are considered to be different.)

• What is the largest possible value for S?

• How many different assignments of vertex values give this maximum?

The solution will be posted shortly.

Back to the Archives