Port-and-Sweep Solitaire

Port-and-Sweep Solitaire is a puzzle game played with counters on a board with a square grid. The fundamental rule of PaSS is:

• Each square may hold up to 2 counters — no more.

There are two kinds of moves, which you can play in any direction (left, right, up or down):

• Port move:
  Take two counters from a square, then add one to another square two steps away (horizontally or vertically). Try a port down from d4 to d6 in Figure 1 (you can also click and drag from d4 to d6 on the figure itself).
• Sweep move:
  Take one counter each from three squares in a row, then add two to a square at the end of the row. If you've already made the d4→d6 port move in Figure 1, you can follow it up with a sweep right from a4 to d4 and then a sweep left from f4 to c4.

Each move leaves one less counter on the board. The challenge in most port-and-sweep puzzles is to play until just one counter remains. Jump right in and try to finish the difficult puzzle in Figure 1 if you like, or read on for a gentler introduction (and more puzzles, of varying difficulty).

The puzzle in Figure 2 can be solved in six moves. You can make the moves yourself by clicking and dragging on the figure, or click through the steps here in sequence:

1. Sweep e1 → e4 (click on e1, then drag to e4)
2. Port e4 → c4
3. Sweep d4 → a4
4. Sweep a5 → a2
5. Port a2 → a4
6. Port a4 → c4 leaving one counter at c4, done!

If things go awry, or you want to start over, click the reset button next to the board.

The port and sweep moves can take some getting used to (especially if you're used to ordinary peg solitaire). Figure 3 offers four relatively forgiving problems on the 5×5 square board. Hopefully you won't get stuck for too long on any of them, but they illustrate some typical maneuvers.

Each move has source squares (the squares you're taking counters away from), and a target square (the square you're adding to). To clarify the rules a little further:

• The source for a port move must have 2 counters, and the target must have 0 or 1 (so that there's room to add one more).
• The "in-between" square for a port is completely unaffected by the move, and may have any number of counters.
• Some boards (Figure 7) have gaps (missing squares), and you can port across a gap in the board. "Port" is short for "teleport!"
• The target for a sweep must be empty, to take two new counters.
• All three source squares for a sweep must have counters, but they can be 1s or 2s, and they don't all have to be the same.

Try both possibilities for the first move in the upper-right puzzle. One solution uses two sweeps; the other uses just one. PaSS puzzles typically have a large number of possible solutions, and looking for a solution that ends with a long sequence of ports can be an interesting extra challenge.

The 5×5 square is just about the smallest board that provides for interesting gameplay, but puzzles of a dozen or so moves on this board already make pretty good brain-teasers. The four puzzles in Figure 4 all finish with a single counter in the center of the board, though they exhibit a variety of symmetries in the initial arrangement of their counters. Despite the modest size, I think you'll find them satisfying to solve!

PaSS works well on a variety of board shapes. Figure 5 poses a few problems on the 25-square diamond. They don't all finish in the center, and that can make it harder to visualize a solution. That's where mathematics can help out: with some simple arithmetic, it is easy to deduce where the final counter could end up in these puzzles, even before you solve them. More about the calculations later, though!

I have a separate page full of selected puzzles, which includes many more 12-move puzzles to play on the 5×5 and 25-square diamond boards.

The cross-shaped board in Figure 1, with 33 squares, is called the English board in the peg solitaire literature. George Bell's A Fresh Look at Peg Solitaire gives theoretical reasons behind the popularity of the English board for peg solitaire. Surprisingly, one of the best features holds true for PaSS, too: All (-1)-complement problems on the English board are solvable, no matter where the initial 2 is! They are all rather difficult to solve purely by trial and error; I think the "d2" problem is the most difficult of them.

Figure 6 shows a complement problem on a 6×6 board with the corners removed. I've taken to calling this the German board. Like the English board, all (-1)-complement problems on the German board are solvable. These are much easier than the English problems.

In Figure 7, we have a central complement problem on a 7×7 board with four interior squares missing. I call this the Swiss board (because of the holes). All (-1)-complement problems on the Swiss board are solvable, and challenging. Don't forget that you can port across the gaps!

For brevity, I've included only one complement problem here for each of the English, German, and Swiss boards, but all the possibilities are playable on the big puzzle collection page.

PaSS actually has two kinds of complements; there are (+1) as well as (-1) complement problems. In practice, the (+1) problems, although they look pretty, tend to be less interesting to play, so I haven't included any examples here. More on those later, perhaps.