Solves puzzle from Prequel Adventure comic (Mar 04, 2014 http://www.prequeladventure.com/2014/03/aggy-extrapolate/ ). Sample output: https://gist.github.com/YellowAfterlife/9369636

Main.hx

import flash.display.Bitmap;
import flash.display.BitmapData;
import flash.display.Graphics;
import flash.display.Sprite;
import flash.events.KeyboardEvent;
import flash.events.TimerEvent;
import flash.geom.Rectangle;
import flash.Lib;
import flash.text.TextField;
import flash.utils.Timer;
import openfl.Assets;
//
class Board {
	public var pegs:Array<Array<Bool>>;
	public var x1:Int = -1;
	public var y1:Int = -1;
	public var x2:Int = -1;
	public var y2:Int = -1;
	public function new() {
		var o = true, _ = false;
		pegs = [
			    [o],
			   [o,o],
			  [o,_,o],
			 [o,o,o,o],
			[o,o,o,o,o]
		];
	}
	public function clone(y1:Int, x1:Int, y2:Int, x2:Int):Board {
		var r = new Board();
		for (i in 0 ... pegs.length) {
			var m = pegs[i], q = r.pegs[i];
			for (j in 0 ... m.length) q[j] = m[j];
		}
		r.x1 = x1; r.y1 = y1;
		r.x2 = x2; r.y2 = y2;
		return r;
	}
	public function solve(m:Array<Board>):Bool {
		var n:Int = pegs.length, l:Int, r:Board, c:Int = count();
		m.push(this);
		if (c == 1) return true;
		for (i in 0 ... n) {
			l = pegs[i].length;
			for (j in 0 ... l) if (pegs[i][j]) {
				// up-left:
				if (i >= 2 && j >= 2 && pegs[i - 1][j - 1] && !pegs[i - 2][j - 2]) {
					r = clone(i, j, i - 2, j - 2);
					r.pegs[i][j] = false;
					r.pegs[i - 1][j - 1] = false;
					r.pegs[i - 2][j - 2] = true;
					if (r.solve(m)) return true;
				}
				// up-right:
				if (i >= 2 && j < i - 1 && pegs[i - 1][j] && !pegs[i - 2][j]) {
					r = clone(i, j, i - 2, j);
					r.pegs[i][j] = false;
					r.pegs[i - 1][j] = false;
					r.pegs[i - 2][j] = true;
					if (r.solve(m)) return true;
				}
				// down-left:
				if (i + 2 < n && pegs[i + 1][j] && !pegs[i + 2][j]) {
					r = clone(i, j, i + 2, j);
					r.pegs[i][j] = false;
					r.pegs[i + 1][j] = false;
					r.pegs[i + 2][j] = true;
					if (r.solve(m)) return true;
				}
				// down-right:
				if (i + 2 < n && pegs[i + 1][j + 1] && !pegs[i + 2][j + 2]) {
					r = clone(i, j, i + 2, j + 2);
					r.pegs[i][j] = false;
					r.pegs[i + 1][j + 1] = false;
					r.pegs[i + 2][j + 2] = true;
					if (r.solve(m)) return true;
				}
				// left:
				if (j >= 2 && pegs[i][j - 1] && !pegs[i][j - 2]) {
					r = clone(i, j, i, j - 2);
					r.pegs[i][j] = false;
					r.pegs[i][j - 1] = false;
					r.pegs[i][j - 2] = true;
					if (r.solve(m)) return true;
				}
				// left:
				if (j + 2 < l && pegs[i][j + 1] && !pegs[i][j + 2]) {
					r = clone(i, j, i, j + 2);
					r.pegs[i][j] = false;
					r.pegs[i][j + 1] = false;
					r.pegs[i][j + 2] = true;
					if (r.solve(m)) return true;
				}
			}
		}
		m.pop();
		return false;
	}
	public function toString():String {
		var r = "", n:Int = pegs.length, m:Array<Bool>;
		for (i in 0 ... n) {
			r += StringTools.rpad("\n", " ", n - i);
			m = pegs[i];
			for (j in 0 ... m.length) r += m[j]
				? ((j == x2 && i == y2) ? "x " : "o ")
				: ((j == x1 && i == y1) ? "* " : ". ");
		}
		return r;
	}
	public function count():Int {
		var r:Int = 0;
		for (m in pegs) for (v in m) if (v) r++;
		return r;
	}
}
class Main {
	public function new() {
		var board = new Board();
		var moves:Array<Board> = [];
		board.solve(moves);
		for (i in 0 ... moves.length) {
			Lib.trace("Turn " + (i + 1) + ": " + moves[i]);
		}
	}
	@:keep static function main() new Main();
}

cripes man, really, really cool, but how come you came up with that in just half an hour... frack I really need to work on my skills! hahaha