Learn to Embrace Your Destiny by Playing Sprouts

Swirly lines of fate

Sprouts is another simple pencil-and-paper game (like Tic-Tac-Toe and Dots-and-Boxes) that is played by children the world over. The board starts with some number of dots, with two starting dots being the most basic form of the game. The players then alternate turns drawing linking existing dots by drawing a dot and two lines extending from that dot, either to two previously existing dots or to an existing dot and the new dot itself. For example, on the starting two-dot board, the two possible moves would be :

Option 1

Option 2

Of course, you could play the self-looping move to connect to either dot, but the resulting position is rotationally identical, i.e. it has the exact same effect on the game, a concept that we introduced in our discussion of  Tic-Tac-Toe.

The moves must conform to the following rules:

1. No line may cross another line.

2. The new dot must split the new line into two segments (the new line must extend from two sides of the new dot).

3. Each dot is limited to 3 lines extending from it. This includes each line that touches the dot, regardless of whether the line originated at the same dot. In either initial move seen above, only one more line may connect to the new dot.

The winner of the game is the one who plays last, that is, the player who makes it impossible for his or her opponent to make a legal move.

Strategy

The core concept of sprouts is the remaining number of lines that can be drawn to dots, or remaining “lives” of dots. Since each player is trying to play the last move, each is trying to arrange the board so that they can be the one to exhaust the remaining lives of the dots.

Choosing option 1 of the initial moves already illustrated, we can see that there are 5 lives remaining after the first move:

5 remaining lives of dots

The second player here gets to make a choice how to arrange the remaining 4 lives:

4 remaining

Also 4 remaining, but arranged differently

While the number of lives remaining here doesn’t change between the two options, the arrangement is significant, at least asthetically. In the first option, it is still possible to draw a dot which connects to itself, as follows :

Loop

Whereas the only possible moves in the second option are connecting two dots in some rotationally identical way o,r connecting with a dot in the center.

Rotationally identical option

Center crossing option

Any option leaves the number of lives at three:

but this is where the strategies diverge. Examining only the first option:

We notice that with any legal move, the next player to play loses the game.  Any legal move allows the final unconnected dots to be connected.

   =       =  

With the second option:

there are two possible moves:

The first of these allows the opposing player to win by connecting the two outer dots, the second option wins on the spot.

The third option:

Allows only one move, connecting any two live dots:

Which loses when the other player connects the final available dots.

If you can see a few moves ahead, (and eventually begin to recognize patterns within the game to sidestep the necessity of brute-force calculation) you can see down a few of the branching options that present themselves, hopefully to a road that brings you to victory.

Passing the move

Since the number of lives of dots left is such an important number to keep in mind in sprouts, one may easily fall into the trap of considering ONLY that number, but there are other considerations. For example, in this position that we looked at earlier:

Stranded dots

Here there are 2 lives left, but no legal moves. This is because the last played dot is stranded, that is, not connected to three lines but unable to legally be played. Stranding dots is actually the only method of passing the move in sprouts. If you master the use of stranding dots and correctly calculating your opponent’s options for doing so, you master sprouts.

Extremely Advanced Strategy and Theory

Because sprouts is a combinatorial game, that is, a two-player game of perfect information with no random element (it was in fact invented by mathematicians in the 1960s), it has been solved for initial boards containing 44 or less dots  as well as specific arrangements of 46, 47 and 53-dot boards.

However, the solutions to the game may not be useful to those simply wishing to play a good game of sprouts. For those players among you, I wish you hours of fun drawing lines and dots in large and outrageous arrangements. However, if you want to win, the only way to gain an advantage in sprouts after mastering the use of stranded dots and the calculation of opportunities to use them is to memorize the following table (borrowed from the solution page noted earlier) and note the result listed for the first player. If it is a loss for the first player, play second. If it is a win for the first player, play first.

Bonus Section:

If you are further interested in the theory behind the game, here are some links that may interest you:

Winning Ways

Ongoing Solution Site

Combinatorial Game Theory

In our zeal toward covering as many games as possible, there will be several games that seem like foregone conclusions in one way or another. Many games operate this way, though sprouts is a game of sufficient visual complexity and equivalent decisions that it still produces a rich experience for a certain number of games.  Even though the strategy of the game may be largely irrelevant at a high level of play, I still encourage you to try this game. It produces some very cool images.

Thanks for reading, and remember to comment if there is a particular game you would like to see covered on Game Gaming.

Category : Games
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