I avoided using a computer - it's more fun as a logic puzzle. The trick is to start with 3-18-1-19-2, which has to be there somewhere. Then, name the points x (centre), y_i (degree 2 node on spokes), z_i (degree 3 node on spokes), w_i (degree 2 node on rim). Then create equations for the sum of the outer segments, the sum of all segments, and the sum of all nodes. These show x must be even, which means it can't be 1 or 3. So either 2 is at the centre, or 4 (since either 2-17-3 or 1-17-4 must appear). Then you need to take advantage of the fact that there are 10 odd numbers and the number of odd numbers in y_i and z_i must match. You also need to use the fact around any triangle there must be either 0 or 2 odd numbers in the middle of the edges. This ultimately leads to ruling out 2 at the centre, and eventually you find the result with 4 at the centre.

What I really want to know... is how did the question setter know there would be a solution?

I avoided using a computer - it's more fun as a logic puzzle. The trick is to start with 3-18-1-19-2, which has to be there somewhere. Then, name the points x (centre), y_i (degree 2 node on spokes), z_i (degree 3 node on spokes), w_i (degree 2 node on rim). Then create equations for the sum of the outer segments, the sum of all segments, and the sum of all nodes. These show x must be even, which means it can't be 1 or 3. So either 2 is at the centre, or 4 (since either 2-17-3 or 1-17-4 must appear). Then you need to take advantage of the fact that there are 10 odd numbers and the number of odd numbers in y_i and z_i must match. You also need to use the fact around any triangle there must be either 0 or 2 odd numbers in the middle of the edges. This ultimately leads to ruling out 2 at the centre, and eventually you find the result with 4 at the centre.

What I really want to know... is how did the question setter know there would be a solution?