6 Exploring Peg Solitaire
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\(\def\upEta{\unicode{x0397}}\)
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\(\def\upNu{\unicode{x039D}}\)
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6.2 Backtracking¶
Now that we know how to represent boards and analyze possible moves, we can turn to the problem of assembling these moves into solutions that lead from INIT_BOARD to FINAL_BOARD . The simplest algorithm for systematically finding such
solutions is based on a recursivebacktracking
Backtracking is a general procedure for generating all possible arrangements of certain objects — in our case, the various sequences of moves that can be performed in peg solitaire. Every backtrackingcurrent , the idea is to use recursion to exhaustively explore all possible solutions that start with current . In pseudocode, this looks as follows:
backtrack(current) {
if (current == desired solution) {
print("Solution found: " + current)
} else {
for (next : states reachable from current)
backtrack(next)
}
}
The use of recursion ensures that returning from the function automatically “backtracks” to the last partial solution that needs to be explored further.
It’s easy to adapt this general scheme to our problem of generating all possible solutions of peg solitairecurrent is now the current state of the board, and the states that are reachable from current are the boards that are
obtained by making a single jump:
solveBoard(current) {
if (current == FINAL_BOARD) {
print("Solution found")
} else {
for (next : boards reachable from current in a single jump)
solveBoard(next)
}
}
As illustrated in Fig. 6.5 , this backtracking
Figure 6.5 The game states of peg solitaire can be represented as a tree, with the initial state at the root. Each node’s children consist of the states that can be reached my making a single jump. Gray circles indicate sub-trees that have been omitted for clarity.
Listing 6.5 shows one possible implementation of the backtrackingfor loops to generate the possible jumps: The first one iterates over the
movable pegs and the second one over the reachable holes for each peg. We then try each possible jump recursively. The member variable jumpList stores the current sequence of jumps as a nested list of integers; each entry in this list is a pair of integers (peg , hole )
that indicates which peg was moved to which hole. New entries are added to jumpList before each recursive call to printSolutions() and removed immediately afterward.
Listing 6.5
private final List<List<Integer>> jumpList = new ArrayList<>();
public void printSolutions(long pegs) {
if (pegs == PegBoard.FINAL_PEGS) {
System.out.println(jumpList);
return;
}
for (int movable : Bits.iterate(PegBoard.movablePegs(pegs))) {
long reachable = PegBoard.reachableHoles(pegs, Bits.bit(movable));
for (int hole : Bits.iterate(reachable)) {
jumpList.add(List.of(movable, hole));
printSolutions(PegBoard.jump(pegs, movable, hole));
jumpList.remove(jumpList.size() - 1);
}
}
}
We can now solve peg solitaireprintSolutions() with INIT_PEGS as the starting configuration:
new PegSolve().printSolutions(PegBoard.INIT_PEGS);
The program quickly finds the first solution and continues to generate several hundred solutions per second. Here are the first two lines of the output:
[[10, 24], [15, 17], [2, 16], [4, 2], [17, 15], [14, 16], [18, 4],
[20, 18], [23, 9], [2, 16], [21, 23], [23, 9], [25, 11], [4, 18],
[27, 25], [25, 11], [37, 23], [28, 30], [30, 16], [9, 23],
[23, 25], [32, 18], [11, 25], [34, 32], [31, 33], [46, 32],
[25, 39], [44, 46], [46, 32], [33, 31], [38, 24]]
[[10, 24], [15, 17], [2, 16], [4, 2], [17, 15], [14, 16], [18, 4],
[20, 18], [23, 9], [2, 16], [21, 23], [23, 9], [25, 11], [4, 18],
[27, 25], [25, 11], [37, 23], [28, 30], [30, 16], [9, 23],
[23, 25], [32, 18], [11, 25], [34, 32], [31, 33], [46, 32],
[44, 46], [25, 39], [46, 32], [33, 31], [38, 24]]
...
As you can see, the two solutions are almost identical and differ only in the order of the last few jumps. This is a general characteristic of peg solitaire: Once you have found one solution, you can usually generate many more by reordering jumps that are independent of each other.