Tuesday, October 27, 2015

The Four-Color Theorem and The Pumpkin Time-Bomb

"CRAYONS!? We are COLORING? Wait, can I snapchat this????"

Yes, yes you may.
Here are the directions I gave to my Problem-Solving class the other day:

Take any map and color it so that no two adjacent states are the same color. Keep in mind that you want to use as few colors as possible so that coloring the map is not too expensive. It turns out that any map can be colored in at most 4 colors! And that this was the first proof proven using a computer. 

Here is the map I gave them, without telling them that they needed 4 colors...you may want to provide extras.

Here is the quick video I showed them after they colored, that talks about the four-color theorem.

After coloring the map, we drew the graph at the right, where each vertex represents a country and an edge connecting two vertices means those countries are adjacent. Then we colored the vertices and found the "chromatic number," that is, the least number of colors we can use. We needed at least three, as you can see by the red triangle drawn between Bolivia, Paraguay, and Argentina. The chromatic number for this map was 4. (And remember, for any map, this is the most it will ever be.)

Then we went on to scheduling committees, and I showed how you could make each vertex a club and each edge would represent any club that had a member in common, so a conflict in time. The chromatic number, in this case, would be the least number of club meetings required. We also answered some interesting problems that related to the Handshake Problem.

Here are my notes from class that day, that I took from my book. 
All in all, a fun class with a theorem that they have never heard before. And I am sure that they will never forget.
Today, we did the awesome Pumpkin-time-bomb-activity from Mr. Orr. Very fun predictions, and we were only 4 off from what we guessed! I highly recommend that you read his post, check out the data, and also watch the video from Jimmy Fallon.

The explosion took us by surprise so we didn't get a picture, but one student ended up with pumpkin all in her hair! I was amazed at what the rubberband clump looked like at the end! We didn't do anything to it. That's what it looked like after the explosion!

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