Zugzwang is a great word for people who like games, particularly competitive strategy games. German for “compulsion to move”, a player in zugzwang is forced to make a move even there is no good move to make; no move at all would be a better alternative, but because of game rules they are compelled to do something. The word appears in chess literature in the mid 1800’s and in English by 1905. There is a formal definition of this term from a branch of mathematics called combinatorial game theory, but the practical definition is what I will focus on here. In this post, I will take other concepts from game theory and apply them in a practical way to analyze a classic game and make it better (more fun, more exciting, etc.). In particular, I made Floating Four to address a zugzwang problem with its predecessor. So let me describe the game, list some theory, tie the theory into practical results and describe why the new game is better.
The Captain’s Mistress
Many people will recognize this game as “Connect Four” which is a trademark of Milton Bradley since 1974. The original game is much older and is attributed to an unconfirmed story that explorer Captain Cook spent so much time stuck in his cabin playing the game against his fellow travelers naturalist Joseph Banks and botanist Daniel Solander on his voyages that it become known as the Captain’s Mistress. It has spherical tokens and a “gangplank” that is removed to reset the game. Other names for this game include: Four Up, Plot Four, Find Four, Fourplay, Four in a Row and Four in a Line.
The rules are simple. It is a two player game where each player chooses a color and take turns placing tokens of that color on a board. The first player to get four of their tokens a straight line (vertically, horizontally, or diagonally) is the winner. The board of play is suspended vertically, with seven columns and six rows. The tokens are dropped from the top of the board into a column and fall down to the lowest unoccupied row.
Problems with the Game
First, it is not a fair game. Sure, it may seem fair: each player takes turns, each player has the same number of tokens, etc. but “fairness” has a definition when it comes to games. A fair game is one where each player, playing their absolute best, have an equal chance of winning. In 1988, two mathematicians independently proved that is is not a fair game, the first player can be unbeatable with the right strategy. That’s not fair, and not fair can also be not fun when people figure this out.
The more obvious cause of “not funness”, however, is the zugzwang problem of this game. Ever be in a position where if you drop a token in a slot, your opponent will drop his piece right on top and win? You know how you can make another twenty moves filling up the board only to be forced to drop in that column and give them the win? If so, then you know the pain of protracted zugzwang. During the entire time, the player in zugzwang has no hope for an upset against a skilled opponent. What kind fun is that? None, I tell you.
Why Floating Four is Better
I looked at the fairness and zugzwang problems with the original game and asked, “What is the simplest game element we can add that fixes those?” The answer that came up was “gravity.” It turns out you could do the same thing with a physical game just by flipping the board (assuming you close it first, so the pieces don’t spill). But a physical version of the game has other problems, particularly when it comes to resolving ties. Let’s first discuss the new rules and why they are designed the way they are. Then we’ll get to how they fix the fairness and zugzwang problems.
Rule #1. Reversing Gravity. The game starts with tokens entering a column at top of the board and falling down to the first unoccupied row. Players now have the option of forgoing their token drop and reversing gravity instead. This causes all tokens to float (or fall) to their new position at the top edge (or bottom edge) of the board. When this happens, tokens remain in their column and do not change position relative to tokens above or below them. Positions of tokens on either side of them are another matter.
Rule #2. Frequency of reversing gravity. The ability to change whether tokens float or fall is not always available.
- At the start of the game it is disabled until the fifth move.
- After gravity has been flipped, it cannot be flipped again for two more moves.
The hardest part here was actually coming up with the padlocks as a visual idiom to communicate when players could and could not switch gravity.
Rule #3. Resolving ties. If, as a result of a gravity flip, both players have winning runs of four in a row, then the player with the most winning runs is declared the winner. If both players have an equal number of winning runs, then all the tokens involved in winning runs are flagged and removed from play. The remaining tokens then fall (or float) to occupy any vacancies in their columns and gameplay continues. Note. it is possible for multiple rounds of ties to occur.
Reversing gravity had to be limited for two reasons. First, we wanted to guarantee that the game always progresses. If a player flips gravity and another flips it right back, then tokens slosh back and forth but nothing moves forward. So it had to be at least one move. Two seemed to be a good balance, because it forced both players to insert tokens in the new direction before reverting it back. Why five in the beginning? Primarily, we wanted an odd number to give the second player an advantage to offset the advantage the first player enjoys. Five is also perfect to defeat the simplest fork position where player 1 gets three tokens across the bottom edge and player 2 cannot block both sides at the same time.
Resolving ties, and therefore removing tokens from the middle of a column, is the action that lends itself most to a video game as opposed to a physical game. I considered leaving the ties in there and just declaring a win when a player gets more runs than another, but there are problems with such an approach. It is actually harder for people to count runs correctly when there are too many of them. More importantly, a second reversal of gravity would cause previously winning runs to disappear, which is not a fun experience either. Removing the ties seemed the best approach and greatly increased the number of legal configurations of tokens that can appear in normal gameplay. There is also potential for greater imbalance. For example: if gravity flips and both players have two winning runs — say light has five in a row and dark has two independent runs of four — then light has only five tokens removed whereas dark has eight removed! This clearly puts dark at a disadvantage later in the game.
Even with all this, I cannot overstate how important it is to test a game… especially kids. I did our first prototype of this game in a day with Python and a simple text interface. We played hundreds of games, and had the AI play itself thousands of games, just to see how much gameplay could vary. There are some arrangements of tokens that effectively prevent the opponent from ever flipping gravity, and ways to disarm that trap. Children especially like to flip gravity as a kind of “panic button” which often gives unexpected, but wonderful results.
I hope you try Floating Four yourself on iTunes and tell me what you think.