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Cartographer

A classic math proof is also a classic math coloring activity. Only four colors needed!

Ages: 4+
Players: 2+
Time: 10-20 Minutes
Type: abstract games
Location: tabletop
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Ages: 4+
Players: 2+
Time: 10-20 Minutes
Type: abstract games
Location: tabletop

Instructions

Imagine you're coloring in a map of the United States. You want the different states to stand out from each other, so you avoid using the same color on states that border. (It's okay if states that only touch at a corner are the same color.) What's the fewest number of colors you would need?

If you'd like, let your math beast give this a try before giving the answer. Here is a map of the United States to print and color in.

Blank map of USA with markers

It turns out you can color in your map using only four colors.

US map colored in
One way to color the US map using only four colors. (Or, surprising electoral results?)

Remember, regions that border each other can't be the same color, but if two regions only touch at a corner, they can be the same color. (So in the map above, both Arizona and Colorado can be red, even though they touch at a corner.)

What's even cooler is that you can color any map using only four colors. This fact is called, unsurprisingly, the Four Color Map Theorem. Try making your own maps and coloring them.

drawing of map on paper
drawing of map on paper
drawing of map on paper
drawing of map on paper
drawing of map on paper

Try to use fewer than four colors. Some maps only need three, or even two colors.

drawing of map on paper
drawing of map on paper colored in

There are many ways to make a game of this. Try drawing a map that "breaks" the Four Color Map Theorem (that is to say, that would require at least five colors to fill in). It's an impossible but fun challenge. Or, draw maps and give them to each other to fill in.

Or, make it a game. Start with any blank (not colored in) map.

drawing of map on paper with markers beside

Take turns using a color to fill in any region of the map.

Coloring in the first part of the map
Coloring in the second part of the map

Try to force your opponent to need a fifth color. In the picture below, using any of the four original colors would cause two regions of the same color to border, which isn't allowed.

Map mostly colored in

The next player won't be able to fill in a region without using a fifth color, so they lose.

Two regions left blank on map

See Variations below for other possible games.

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

Collaborative Play:

This variation works for two or more players, and is great for younger players. Start with a blank map, as in the game described above. Take turns coloring in a region. Instead of trying to force your opponent to need a fifth color, play collaboratively. Your goal is to be able to color in the whole map.

Pass The Map:

This variation works for 2-4 players. You'll need 4 markers total, as well as a piece of paper for each player. You'll also need a pencil or a fifth marker to draw maps with. Each player starts by drawing a map on their own piece of paper. All of these maps need to have the same number of regions. 12 regions is a good starting number. (For a younger player's map, an adult may need to look and count the regions, and clarify any ambiguities, such as whether two regions border or not.)

Next, distribute the markers. For a 2-player game, each player gets 2 markers. For a 4-player game, each player gets 1 marker. For a 3-player game, each player gets 1 marker, and the extra marker is left unused. A player keeps their marker (or markers) throughout the whole game.

To play, color one region on your map. Once everyone has done so, pass your map to the left, and then color one region on the map you were given. For a 2-player game, you can choose which color to use each time. For a 3- or 4-player game, you must use your same color each time. Continue passing maps, then coloring, passing, then coloring. When a player isn't able to color a region (without bordering a region of the same color), they are out. The map they had when they lost is also out of circulation. The other players continue coloring and passing their maps. The last player to be able to color in a region wins!

Challenges:

Ask your math beast to try drawing the following: a map that has 4 regions, but requires 4 colors to fill in. Or see if they can figure out how to draw a map with as many regions as they like, but that requires only two colors. Here are some possible solutions.

Four regions colored in four colors
A 4-region 4-color map
Pictures colored in two colors
Maps that require only 2 colors

Classroom Tips d

Cartographer can be coordinated with a Social Studies unit involving maps. Or play Cartographer (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

  • See Challenges below for specific things to ask students to try.
  • Can you make a map that only needs 3 colors?
  • Why do you only need one counterexample to prove something is false? (see Learning Notes)
  • What fraction of the total number of regions is each color?
  • Can one color ever cover more than half of the total number of regions?

Alignment with Beast Academy Curriculum

  • 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

Math History:

The question of whether a map could be drawn that would require five colors has been asked since at least the 1850s, when mathematician Francis Guthrie was inspired by coloring in a map of the counties of England. For over a hundred years, no one was able to prove, one way or the other, whether this was in fact impossible. Many tried, and there are numerous instances of people coming up with maps that they believed could not be filled using four colors (only to be shown they were wrong). Finally, in the 1970s, mathematicians Kenneth Appel and Wolfgang Haken used a computer to prove that every map could be colored with four colors. This was the first time a computer was used to prove a mathematical theorem (and it was very controversial at the time!).

Counterexamples:

In many cases, proving something to be false is easier than proving something to be true. Before the Four Color Map Theorem was proven, many people tried to draw a map that would require five colors. If any one of them had managed to succeed, their map would have been a counterexample to the entire Four Color Map idea. You only need one counterexample to show that a statement is wrong.

You can practice this with your child in a game called "Nope." Make a false statement, like "All birds can fly." Then let your child think of a counterexample, to prove you wrong. In this case, they'd say, "Nope. Penguins are birds, and penguins can't fly." Or you could say, "All vegetables are green," and let your child respond, "Nope. Carrots are vegetables, and they're orange." Or try something mathy: "All even numbers end in 2, 4, 6, or 8," to which your child could respond, "Nope. 10 is even, and it ends in a 0."

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Materials
  • paper
  • markers
Learning Goals
  • spatial reasoning
  • wonder
  • strategic thinking
Common Core Standards
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Image of Ms. Q

Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

LEARN MORE

Bring problem-solving to your classroom

Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

LEARN MORE
Image of a BA book