300000

Total number of wins: 4128

(6 – 20): | Cycle Path Random Challenge |

This game is quite new to our website, so this Food For Thought is under construction. One should consider checking back later for more tips.

Definitions + How is this still a "math" game?

The diagrams in this game are based on a field of mathematics known as graph theory!

Basics of graph theory

**Graph**: A graph is a diagram that shows how objects are connected in pairs.

**Node**: The "objects" a graph connects are called nodes. In Spider Web, nodes are the **circles**.

**Edge**: An edge is a connection in the graph between two nodes. In Spider Web, the edges are **lines**.

Walking the trail to the solution

**Walk**: A walk is a sequence of edges, where **each edge starts from the node on which the previous node ended** (2 nodes and the line in between). This could be a person "walking" along the city streets, or a spider "walking" along the strands of the spider web.

**Trail**: A trail is a kind of walk that **never uses the same edge twice**. Hiking "trails" shouldn't repeat lengths, or else the hikers could get lost or bump into each other!

**Eulerian Path**: An Eulerian Path is a kind of trail that **uses every edge in the graph** (exactly once). In Spider Web, this is the objective.

Does the name "Euler" sound familiar?

Fun Fact! Graph theory was invented by mathematician Leonhard Euler.

Euler wrote a paper on the Seven Bridges of Königsberg proving that it was impossible to walk through the city crossing over each bridge exactly once.

Click the map of the problem below to read more about it.

**Cycle**: A cycle is a kind of trail that **ends on the node it started on**. What do you think of when you hear the word "cycle"?

**Eulerian Cycle**: This is **an Eulerian Path that is also a cycle**. Select "Cycle" in the table above and the solution will always be an Eulerian Cycle!

Game Settings Instructions

In the textbox next to **Nodes** you can type in any number between 6-20. This number will be the number of nodes displayed on the graph. The larger the number the more challenging the solution becomes!

Next, select a Type of graph:

**Cycle**: Winning paths will always end on the node they started on.

**Path**: Winning paths will not end on the node they started on.

**Random**: Winning paths may or may not end on the node they started on. You'll need to plan your moves wisely!

**Challenge**: The Caribou team has hand-crafted these especially tricky graphs for those who need more of a Challenge.

Finally, click "Create a New Web" to see your changes to the graph settings!

What node should I pick first?

**Degree**: The degree of a node is the number of edges connected to it.

A graph (undirected – meaning no directions on the edges) contains an Eulerian Cycle when **all of the nodes have an even (0, 2, 4, 6…) degree**. For these, **you may select any node as your starting node** and still win. Just keep in mind that you will have to finish on the node you started from.

A graph contains an Eulerian Path, but **not** an Eulerian Cycle, when **exactly two nodes have an odd (1, 3, 5, 7…) degree** and all other nodes have an even degree. In this case, **you must select a node with an odd degree as your starting node** or you will not be able to win!

Why exactly two nodes?

Think of each move as **"out" of the node you came from and "in" to the node you travel to**. Most nodes will follow this pairing pattern, so they will have an even degree.

When you start the game, the first edge you mark is a move "out" of the starting node. **This remains unpaired as you play the game.**

To make an Eulerian Cycle, the final edge you mark is a move "in" to the starting node. This creates a pair with the first move, meaning **the starting node has an even degree**.

If the graph's Eulerian Path is not a cycle, the final edge you mark is an unpaired move "in" to a node that is not the starting node. The **starting node** remains with an odd degree, and the **ending node** will also have an odd degree.

Don't worry about any other cases. If the number of nodes with an odd degree were not exactly 0 or 2, there would be no Eulerian Paths and no Eulerian Cycles, meaning no way to win. Graphs like that should not appear in this game.

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