2025 Gauss 8 Problem 24

In the diagram, circles are *connected* if they are joined by a line segment. Each circle is filled with one integer so that

- the positive difference between each pair of integers in connected circles is d, and - the sum of the five integers in the circles is 54.

For how many different values of d between 1 and 20 inclusive can the circles be filled in this way?

\textbf{(A)}\ 4\quad \textbf{(B)}\ 12\quad \textbf{(C)}\ 8\quad \textbf{(D)}\ 20\quad \textbf{(E)}\ 16

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.