EULERdem

last revised 4/13/02

PURPOSE:

This demonstration will present information to acquaint the audience with graph theory and the Euler path. It also relates to the graph theory shown in the Beyond Numbers exhibit.

OBJECTIVE:

1. To demonstrate how the use of odd and even numbers can help you solve mathematical problems.
2. To demonstrate how graph theory is different from coordinate graphing.
3. To demonstrate how using graph and networks can be used to help in urban planning.

MATERIALS:

Wipe-off magnetic board with marker
8 magnetic dominoes
10 gold plastic cones
graph for dominos
graph for castle
Vampire cape
name labels for Donor, Vampire
large disposable paper graph
cards that display vocabulary "Euler" and "Europe"
1 certificate for each volunteer
stool for dungeon
rug and foam walls for castle
castle blueprint

OUTLINE:

Lay out mats for castle scene to the left front of the stage. Set up castle pieces on the mats and "cones" in the doors. (Velcro on the blocks go down on the mats. Markings on the block and rug show where they go)

Set up magnetic board with marker on easel to the left side of the stage. Scatter dominoes on it.

Have the paper graph concealed within reach behind the board. As well, the euler/europe signs.

On another easel to the right of the first, place the knob chart. Also have a castle chart on the back of the board.

Have certificate for volunteer.

A.Dominoes Activity:

Invite a volunteer on stage to arrange the dominos. Without looking, tell the volunteer that the end dominoes are green circles and yellow squares.

Introduce the topic of graphing. Clarify that graphing in this demonstration is not graphing cartesian coordinates.

Show how graphing is done for the domino trick. Show how knowing odd and even vertices makes the trick possible.

B. The Count's Castle Activity:

Invite a volunteer onstage to act the part of the Donor who tries to steal all the gold cones and get them back to his headquarters.

The Donor tries to go through every door only once, graph his route (the outside being the Donor's headquarters).

Ask the Donor how the problem could be solved. Make a new door in the dungeon and have the volunteer go through the game again.

Give the volunteer their consolation prize- certificates telling a little about graph theory with a few puzzles about the castle.

DEMO PROCEDURES:

1. Setup

Lay out the mats for the castle scene to the right front (near the stage lighting console) of the stage. Set up the castle pieces on the mats and cones in the doors. (Velcro on the blocks go down on the mats. Markings on the block and rug show where they go)
Set up the magnetic board and marker on its easel to the leftmost side of the stage. Scatter the dominoes on it. Have the paper graph concealed within reach behind the board. As well, put the euler/europe signs. On the front ledge, place the certificate for the volunteer.
On another easel to the right of the first, place the knob chart. On the rightmost wall (under the flag spots) hang up the castle layout.

2. Dominoes

Invite a volunteer on stage. Explain the rules of dominoes, saying how, in this game, you put two dominoes next to each other, with the parts that have the same picture touching, but no more than two shapes touching anywhere. Tell your volunteer to arrange the dominoes. Without looking, tell the volunteer and the audience that the end shapes are green circles and yellow squares.
Explain about even and odd numbers contributing to this trick. Of all the shapes, the only two which are represented in an odd number are the circle and square. Show how each evenly represented shape can be completely paired off, but each of the odds must have one remaining.
Then introduce the topic of graphing. Clarify that graphing in this demonstration is not graphing cartesian coordinates, but instead graphing networks with edges and vertices. Feel free to pull out the paper graph to demonstrate cartesian graphs. Tell how Euler's theory is what we used to do the domino trick. If you want, pull out the cards for Euler (oy-ler) and Europe (oy-rup). Be prepared to face many groans.
Then point out the knob board with the domino shapes. Choose one end of the domino chain and place the rope around that shape. Ask the audience to call out each consecutive shape in the chain as you connect those shapes on the board with the rope. In the end, you will have a network that looks like a pentagram inside an almost?pentagon. Once again show how only the circle and square "vertices" are odd by the number of lines coming from them.

2. The Count's Castle

Invite a volunteer onstage. Tell her that she will act the part of a thief who tries to steal from the vampire's castle. To win, she must get ALL the gold cones and bring them back to her home. However, she must follow three rules: Once picking up a gold cone, she must go through that doorway. She may NOT go through an empty doorway. And everytime she goes out of the castle, she must take the cones to her house.
Before she begins, draw four points on the board, representing the dungeon, kitchen, shop and home. While she tries to get every cone, map out her route, drawing lines between the points. Try not to cross the lines as crossings make the map confusing.
At the end of the game, the volunteer will be stuck in the dungeon. Ask her to return all the cones to the doorways as you explain what happened. Refer to the completed map on the board. Show how the only points in the network that have an odd number of lines coming from them are the home and the dungeon. Remind the audience that Euler's theory says to complete such a network, you must start from an odd and end on an odd. Since the "Thief" started at one odd vertex, the home, she had to end at the other, the dungeon.
Ask the audience how the problem could be solved. If there were an additional door in the dungeon, the thief could have left through it in the last step. Move one of the walls in the dungeon to make a new door and put the extra cone there. Have the volunteer go through the game again. This time, there should be no difficulty in getting all the cones and escaping.
Give the volunteer her consolation prize - a certificate telling a little about graph theory with a few puzzles about the castle.
Explain to the audience that when you added the new door in the dungeon, the two odd vertices became even. Euler said that the restrictions that hold true for a network with two odd vertices don't hold for one with no odd vertices. This new network could be completed with the end point being the same as the starting point.

SAMPLE SCRIPT

Welcome to the Maryland Science Center. Let's begin with what appears to be a "magic trick". I have here a set of dominoes with different shapes on them. The shapes are pink triangles, yellow squares, green circles, white stars, and red hearts.
I would like a volunteer. [Choose a volunteer] When I turn my back, I want you to put these dominoes in a line matching the shapes. [Show a sample of connecting dominoes]. Our special rule is that one domino symbol can only connect with one other symbol. No bridging off of a connection. When you are done, I am going to be able to read your mind and tell you which shapes are at either end of the line without looking.
Now, arrange the dominoes in a row. Audience you may have to help them out. [Tell the rest of the group to say "Ready" when the dominoes are properly in a row.] Is each domino symbol connected to only one other domino symbol? Now come over here where I can see you. (Place hand near volunteers head and pretend to read their mind.) Are the symbols green circles and yellow squares? Let's have a round of applause for our volunteer. [Have the volunteer sit]

This may have appeared to be a trick but it actually involved math. We can figure it out by looking at odd and even numbers. If we look at the green circle, we can see that there are only three of those shapes. Two of them will pair off leaving one without a partner so that one is the odd one out and has to be on an end. If we look at the yellow square, we find the same thing. There are three yellow squares. One yellow square won't have a partner. So the yellow square is also the odd one out. If we look at the other shapes in the middle, they all have partners. They are even numbers. So I fixed it so that a yellow square would be on one end and a green circle would be on the other. I rigged the game.

Well, today we're going to talk about how odd and even numbers can help you solve problems. We will be exploring a branch of mathematics called topology. Topology is simply the study of how things are connected. We can show these connections on a graph.

When I say the word graph, you may think that I mean points graphed on an x and y axis. [Pull out MTV graph] Well, this is not that type of graph. [Throw away MTV graph] The type of graph we are talking about looks like this. (pull out pegboard with the pentagram on it.) This graph involves vertices and lines or edges. [Indicate the board with knobs.] These points are called vertices. A vertex is a point that has lines coming from it joining other points or vertices. Using this sort of graph to solve a problem is called graph theory. Graph theory was thought up by a man named Euler (OIL-er)[Show EULER sign]. Euler was a mathematician from Switzerland. That means that he came from "OY-rup" [Show EUROPE sign...Pause for reaction to corny joke.]

Euler came up with the idea that problems with these graphs can be solved if there are only two odd vertices or all even vertices. These types of networks or graphs are used in urban planning, computer programming and business operations. In urban planning you can look at all this as if all of these lines were one way streets. And all of the knobs are stadiums such as Camden Yards. If you have two streets leading into a stadium, you would need to have two streets leading out of the stadium or you might have traffic problems.

Let's play another game involving blood, greed and. . . math. I'll need a volunteer. [call for volunteer] Here I have a castle. There are three rooms in the castle. There is the kitchen, shop and a dungeon. [Indicate the rooms on the map] In the castle is an evil vampire. But, you live in this little tiny home, a hovel. [put volunteer in the hovel] The vampire is so rich he uses gold as doorstops, and you have hardly anything. You decide the the vampire has plenty of gold so you are going to steal some of his gold doorstops.

WARNING: We don't suggest stealing of any kind unless under the strict supervision of the Maryland Science Center Staff.

Now when you go to take the gold, you can take gold out of the doorway, but once you have removed the gold doorstops, the door locks behind you. If you can get out of the castle with all of the gold doorstops, I will give you a fifty dollar gift certificate for the Science Store. But, I really don't want you to win this game because the money will come out of my paycheck.

Now I will try to make a graph of all of the donor's moves. Begin and good luck! [Draw the vertices on the board representing the rooms. As the volunteer passes through the rooms, draw a line contecting to each vertex . The donor will always end up in the dungeon.]

You may suspect that I rigged this game and that there was NO CHANCE of the Donor ever getting the gold. You're right. I rigged it using mathematics because I didn't want to loose fifty dollars from my paycheck. The same way that the domino trick relied on mathematics, I used the same principals here. Let's look at the graph to see.

This also has to do with odds and evens. For this to work for the donor, they have to be able to go in and out of each room. I can go in/out, in/out of the kitchen. So that would make the kitchen even. The donor can go in/out, in/out of the shop making the shop even. The donor can go in/out, in/out and in the dungeon. That doesn't work!! It makes the dungeon odd. The donor will get stuck. So how can we fix it so the donor can win? [Solicit answers from the audience. Put another door in the dungeon.] You see, little did our volunteer know, but in the dungeon there is a secret passage behind this wall. [Remove wall piece to create a new door] So now we have another door in the dungeon. Let's try it again to make sure it works, but this time with no money involved. [Run the game again.] If there are only two odd vertices, the problem will always work in my favor. The first time we played the game, the dungeon and the hovel were odd and the other rooms were even. The second time we played the game, all of the rooms were even. Meaning that I would never win the game; therefore giving my volunteer the 50 dollar gift certificate. This was the same thing in the domino trick.

To thank our volunteer, we have a consolation prize. [Give our volunteer the certificate. You may also give the vampire a prize if you wish] Let's give our volunteer a round of applause.

Thank you all for coming to the Maryland Science Center. I hope you enjoyed this demonstration on Vampires and Math. And I hope you saw that math can be more than you expect. Something as simple as even and odd numbers can help you through incredible adventures, adventures BEYOND NUMBERS!

BIBLIOGRAPHY:
Goldstein, Eddie. The Magical Math Show Teacher's Guide.; 1983
Reimer, Luetta and Wilbert. Euler ? The Bridge to Topology. AIMS, January 1991