Google didn't come up with a result: https://www.google.com/search?q=cracker+barrel+peg+game+solvability+in+higher+dimensions
The Cracker Barrel Peg Game consists of a triangular board with 15 holes for pegs. 14 pegs go in to the holes with one empty. A peg can jump over one other peg into an empty hole to remove the peg jumped over. Solution: https://youtube.com/watch?v=ILKXEnX_YGM
Came to mind when I walked in to a TV show and saw the game on the screen. A friend after I asked in a Discord gave me this: http://mtweb.mtsu.edu/cc2013/Beeler.pdf but it seems to be generalized to two dimensional graphs.
For example, in the third dimension you would be a tetrahedron consisting of layers of 1, 3, 6, 10, and 15 pegs with one peg open. In the fourth dimension you would have tetrahedron layers of size 1, 4, 10, 20, and 35 pegs.
Is this puzzle solvable in higher dimensions? And why?