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.

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."