Summarize this article in one sentence.
Start with position 1 empty. This would be the top point of the triangle. There are other methods of solving the puzzle, but this one is the most common and easiest. In the following videos, grey circles indicate empty holes, red circles indicate pegs you will be moving, and blue holes indicate pegs that are not in play. This basic pattern can be repeated throughout the board, helping you win the game.  Take the peg in position 4 and jump it over the peg in position 2. It will then be in position 1. Use the peg in position 6 to jump position 5. It will now be in position 4. Use the peg in position 1 to jump position 3. It will now be in position 6. These moves will give you the diamond base. As you'll see, the next set of moves will divide the pegs into two groups, one on each side of the triangle.  Move the peg in position 7 to position 2, jumping position 4. Move peg 13 to position 4. Next, move peg 10 to position 8. You should now have two groups of pegs, with 5 pegs on the left side and 3 on the right. This next set of steps will make a curved line across the triangle, dividing it into two parts.  Move peg 2 to position 7. Then take that same peg and move it to position 9. Next, move peg 15 to position 13. These moves will leave you with three final pegs, all in the bottom row.  Move peg 12 into position 14. Next move peg 6 to position 13. You should now just have three pegs left in the bottom row. Move peg 14 to position 12. Then make the last move, peg 11 to position 13. The ending peg should be in the 13th hole. Congratulations! You've just completed the Peg Game.

Summary:
Begin the game. Make the diamond shape. Make two groups of pegs. Separate the triangle in two. Move your last pegs to the bottom row. Complete the last moves. Win the game.