Math - Worm Holes

WebQuest for Wormholes

The
WormHoles
of
Jargolturus

A WebQuest for Future Mathematicians

&

Those Seeking Glory in Trekdom

Drag the Mouse — Save the Worlds!

Introduction

In a galaxy far far away, the planetary alliance of the Jargolturians distributes its most precious cargo, Jargolium, through a network of wormholes. Jargolium is a complex compound of wormhole dust, collectible only when wormholes evaporate. Without Jargolium, the life forms of Jargolturus would soon become extinct.

The only way known to evaporate wormholes is to fly through each with a Jargolium particle collector, scooping up the dust as you fly by. But you must remember, the wormholes crumble and vanish as you fly through them, so there is no going back the way you came.

After distributing the Jargolium among as many Jargolturians as possible, the planetary alliance must patiently wait for the next wormhole configuration to emerge. New configurations occur during each Syzygy of the Seven Suns. In the time of the last syzygy, the Jargolturan wormholes connected seven planets: Arxon, Betelzub, Cantork, Diophonus, Elgarq, Fyjyron, and Galpo. Each of these planets is at least 10 light years from its nearest neighbor, but via wormholes a starship may visit them all while the syzygy is still in effect. At present, these planets are connected as seen in the diagram above. After the next syzygy, expected soon, no one knows which planets will be connected to which. It is imperative that a path be found now to as many planets as possible. To prepare for your starship appointment, you will serve on the Starship OutTrek during the current wormhole configuration. But first you must pass a test simulation.

Task

The Jargolturian alliance must prepare its starfleet for all possible wormhole configurations. Before you are promoted to a starfleet captain, your task is to discover the secrets of wormhole transformations and master the principles of interplanetary path traversal. Below is a sample of the test you must pass to prove you are worthy of starship command. The test consists of 38 networks that represent planetary configurations. The points where lines intersect represent planets. In a network of wormholes, planets are referred to as nodes. Press your mouse button and drag it through the wormholes without taking your finger off the mouse button. Evaporate as many wormholes as you can. Some you will not evaporate completely without retracing some parts of the path.

These are your objectives:

Find a path through each network, if there is one, that will return you to your starting point. A path that returns you back to where you started is called a circuit.
If a circuit is not possible, find a path that will travel you through as many wormholes as possible.
Find all the paths, if there are any, that will visit all the planetary nodes exactly once.

Remember: a wormhole disappears while you fly through it, so you cannot return to any planet via that wormhole ever again.

Interplanetary Networks Sample Test

Process

Observe closely the 38 networks in the diagram above. Below are questions about these networks you need to answer to make your task easier. Think hard—then think some more:

Which paths can you take to travel each wormhole one time only?
Is there more than one path possible?
Which paths will take you back to the planet where you began?
Is there more than one planet where you could start?
If not, what other lines could you add or delete to return to the planet where you began?
How many ways can you visit each planet exactly once?
You can answer these questions by traveling all the possible paths in each network.
However, the Interplanetary Network Table below may help you find a faster way:
1. In the ODDS column, list how many planetary nodes in each network connect an odd number of lines.
2. In the EVENS column, list how many planetary nodes in each network connect an even number of lines.
3. In the PATH? column, if the network contains a path enter YES, otherwise enter NO.
4. In the CIRCUIT? column, if a network contains a circuit enter YES, otherwise enter NO.
On the basis of the information in your table, does a network contain a path if...
1. all the nodes are even?
2. all the nodes are odd?
3. it has more even nodes than odd nodes?
4. it has one odd node?
5. it has two odd nodes?
6. it has three odd nodes?
7. List the networks that contain paths.
8. List the networks that contain circuits.
9. Write a rule for network traversal in 20 words or less.
10. Write a rule for network circuitry in 20 words or less.

Interplanetary Network Table

Evaluation

You will be rewarded for your endeavors with an appropriate rank in the starfleet:

Jargolium Collector

Midshipman Apprentice

Starship Captain

Starfleet Admiral

Score

Mission Objectives Completed

Answered Questions
1, 2, 3

Answered Questions
4, 5, 6

Answered Questions
7, 8, 9

Answered All 10 Exam Questions

Conclusion

In solving this problem and answering the questions, you will have learned the basics of Graph Theory and Combinatorial Topology, which are essential to constructing the networks we all live in such as the internet, interstates, cell phones, cable TV, and Starbucks.

To explore further the awesome topics of Graph Theory and Combinatorial Topology, you will need to uncover knowledge from the distant past on an ancient planet called Ürq in another galaxy far far away. Visit the links below and expand your universe. Coolness...

What other applications can you think of for what you have learned about graphs and wormholes?

Credits & References

Mathematics: A Human Endeavor by Harold R. Jacobs
The Heart of Mathematics: Invitation To Effective Thinking
by Edward B. Burger & Michael Starbird
Discrete Mathematics & Its Applications by Kenneth H. Rosen

HEY!
What's all this Trekkie stuff?
Why not a practical example like this:

Suppose you got a contract with the city to pave the streets of a new subdivision;
but you had to find a path so you could lay out the asphalt without driving over the same street twice.
Why?
Because, Dopey, it takes a day or two for the asphalt to dry, that's why.
You wouldn't want your heavy equipment to get stuck in all that muck now would you?

Worm Holes