Solution: The Penny Puzzle
One solution to The Penny Puzzle is shown below. The numbers denote the starting and ending spaces of the penny moved. In each case the penny in the space between is removed. For example, in the first jump from space 1 to space 6, the penny in space 3 is removed. Here is one solution: 1-6, 10-3, 4-1, 1-6, 8-3, 3-10, 10-8, 7-9. This series of moves leaves one penny in space nine.
The extension question asked if it was possible to solve the puzzle when any of the other spaces (instead of space 6) were left empty at the beginning. If space 5 (the middle space) is left empty there are no possible moves. I was able to solve the puzzle when spaces 2,3,4,6,8, or 9 were empty to begin with, but not starting with 1, 7, or 10 (the corners) empty.