All Activities U Sprouts

# Sprouts

Connect dots with loopy, curving vines in this classic math game. Draw the last vine to win.

Ages: 5+
Players: 2
Time: Under 10 Minutes
Type: abstract games
Location: tabletop
Ages: 5+
Players: 2
Time: Under 10 Minutes
Type: abstract games
Location: tabletop

## Instructions

Draw 3 dots on a sheet of paper.

Now take turns. On your turn, draw a "vine" that starts and ends on a dot. Every time you draw a vine, add a new dot somewhere along the new vine. It's okay for a vine to start and end at the same dot. In the picture below, one vine has been drawn connecting two dots (with a dot added in the middle) and another vine was drawn as a loop, connecting back to the same dot it started from (and also with a dot drawn in the middle).

There are two rules:

1. Vines can't cross other vines.
2. Dots can have no more than 3 vines connected to them. (Once a dot has three vines, it's called "dead." Dots with fewer than three vines are called "live.")

The last player to draw a vine wins!

This simple game was invented by two mathematicians, John Conway and Michael Paterson, in the 1960s. They developed the rules by playing around with different variations until they found something fun and surprising. (A lot of interesting math starts with just "playing around!") Sprouts is a classic favorite of math beasts.

Don't forget: it's Beast Academy Playground, not Beast Academy Study Hall. Change the rules, be silly, make mistakes, and try again. The Variations and Learning Notes are here for you if you want to dive deeper, but not all of them apply to learners of every age. The most important thing is to have fun.

## Variations d

### 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.

## Classroom Tips d

Play Sprouts (or any of our abstract strategy games) after a test or quiz. Once students know these games, they can play them together in pairs or groups if they finish a task early.

Discussion Questions

• Can you figure out who will win before the game is over?
• Does it matter who makes the first move?
• Is there a minimum/maximum number of moves before someone wins?
• How does the number of dots you start with determine the number of connections?

• Level 2, Chapter 12: Problem Solving
• Level 4, Chapter 6: Logic

See Variations and Learning Notes for more ideas on how to adapt this activity and incorporate it into your classroom.

## Learning Notes d

### 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?

## What do you think of this activity?

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Materials
• paper
• pencil
Learning Goals
• strategic thinking
• spatial reasoning
Common Core Standards
• MP1
• MP2
• MP7

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