Seedlings:

For younger players, color a dot red once it becomes dead (or circle it if you don't have another color available). This makes it easier to keep track of where vines can still be drawn. In the examples below, the dots that are colored red are dead. No more vines can be drawn on either board.

More Dots:

Start the game with four or more dots.

Brussels Sprouts:

In this version, start with two plus signs (+) instead of dots. These +'s can have as many as four vines sprout from them (instead of the three-vine limit in the regular version of Sprouts). Think of each + as having four free ends from which vines can be drawn. Each time you draw a vine, draw a short slash across it (rather than a dot, as in the regular version) from which two more vines can sprout. The last player who can draw a vine wins.

Here is a completed game. Normally both players would use the same color, but here Player 1 uses orange and Player 2 uses blue so the game is easier to follow:

Spoiler: This version always finishes in exactly 8 moves, with a win for Player 2! Players can't affect the outcome regardless of what moves they make! Once you know this, the game isn't too fun. But letting your child figure it out for themselves could be fun.

Exploring a Simpler Game:

One way to understand a game is to first explore simpler versions. After playing several rounds of regular Sprouts, try playing a variation that starts with fewer dots.

Though it isn't much fun as a game, try a couple of rounds of 1-dot Sprouts to see how it plays out. (Player 2 wins on their first move, every time.) Then try 2-dot Sprouts. Can your math beast determine the longest possible game that can be played? Or the shortest? (A 2-dot game always lasts 4-5 moves.) Can they figure out which player can guarantee a win on their second move? (Player 2.)

Game Theory:

Another way to understand a game is to keep track of information as you play, then look for any patterns. Older players can do this by creating a chart and recording data about each game they play in a new row. What information should they record? Let them brainstorm what might be relevant. Here are some ideas: the winner (Player 1 or Player 2), the number of dots, the number of live dots, the number of dead dots, the number of vines drawn, how many turns the game lasted.

Eventually, ask if there are any patterns in the data. Guide them toward noticing that if the number of live dots at the end of a game is even, Player 1 wins. If the number of live dots is odd, Player 2 wins. This can be helpful in developing a winning strategy. If you are Player 1, for example, try to create a situation in which an even number of live dots will be left, unable to be connected with a vine.

It's also interesting to notice that the number of total dots at the end of a game plus the number of live dots is always equal to 12. Consider that each dot has 3 "connections" (places where vines can attach) and that there are 3 dots and 9 connections to begin with, for a total of 12. On each turn, 2 connections are lost (since a vine is drawn that uses up a connection at each end) and 1 is added (since a new dot drawn along a vine has only one connection left open). This means the number of available connections decreases by 1 each turn, just as the number of total dots increases by 1. In other words, the number of dots plus the number of connections equals 12 throughout the game.

At the end of a game, each remaining connection corresponds with a remaining live dot. (If a dot had two or more connections, the game wouldn't be over since a vine could begin and end at the dot. This means every remaining live dot must have exactly 1 connection.) So the number of total dots plus the number of live dots will also always be 12. Can your math beast use this to understand why an even number of live dots results in a win for Player 1, and an odd number for Player 2?

How Many Moves?:

Your child may notice that a game of 3-dot Sprouts always concludes in at most 8 moves. The shortest possible game lasts 6 moves. Challenge them to draw such a game. This can be done collaboratively or as a solitaire puzzle. How many different games can they find that end in only 6 moves?