# Problem Solving

## CONTENTS OF CURRICULUM UNIT 80.07.08

## Topology

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## 6. COLORING REGIONS

- a) Provide each student with a copy of figure 33 and add 5 more networks. Ask them to color it so that regions with a common arc have different colors. Try to use as few colors as possible. State the number of colors you need in each case
- b) Draw networks of your own which need: (i) Only two colors. (ii) At least three colors. (iii) At least four colors. Can your neighbor color any of your networks with fewer colors than you have used?
- c) Can you find a map that needs at least five colors?
- d) Draw a map with eight regions which needs only three colors.

### EXERCISE D

Note: In Figure 26 Network 1: A is an even node because 4 arcs meet at A and B is also an even node. In Network 2: A, B, C, and D are odd nodes because 3 arcs meet at each node.

- 1. Provide each student with a copy of figure 34 and add 5 more networks. Ask them to color them using as few colors as possible.
- 2. Design repeating patterns of your own which are suitable for: (a) a kitchen floor, (b) a book jacket, (c) a wallpaper. Color them using as few colors as Possible .

### EXERCISE E (MISCELLANEOUS)

(figure available in print form)

- 1. Figure 35 shows a plan view of a small cottage. Can you start at A and walk through every door of the cottage exactly once? SUGGESTION: Make a topological drawing by making each room into a NODE and each door into an ARC. Can you start anywhere except at A and succeed in traversing the network?
- 2. The Koenigsberg Bridge Problem: In 1737 the Swiss mathematician Leonard Euler was working at the court of Catherine the Great of Russia. He was asked to solve the problem of the bridges of Koenigsberg. This town is now found on maps under the Russian name of Kaliningrad. There are two islands in the River Pregel which runs through the town. Seven bridges cross the river as shown in figure 36. The problem was this: Is it possible to take a walk which crosses each of the bridges once and once only? Euler solved the problem by changing it into a problem about the nodes of a network. See if you can solve the problem.
- 3. THE COLOR GAME: This is a game for two players. The first player draws a region. The second player colors it and draws a new region. The first player colors this region and adds a third. The game continues until one of the players is forced to use a fifth color; this player is the loser. Play this game with a neighbor.
- 4. The diagram 37 shows a simple ground floor plan of a house. Could you walk through this house so that you walked through each door once and once only? SUGGESTION: Make a topological drawing by making each room into a NODE and each door into an ARC.
- 5. GAME CALLED SPROUTS: This game was invented by two mathematicians at Cambridge University in England. This is a game for two players. Make three dots anywhere on a sheet of paper.
- Each player in turn draws a line joining a dot either to itself or to another dot and places a new dot on this line. Some possible opening moves are shown in figure 38.

(figure available in print form)

### RULES OF “SPROUTS”

(i) No line may cross itself, cross another line, or pass through any dot.(ii) No dot may have more than three lines leaving

(iii) When a line is drawn, a new dot must be chosen somewhere on it.

(iv) Each line must join two dots or else one dot to itself.

The last player able to move wins the game. A typical game is shown in figure 39.

A and B are the only dots which do not already have three lines leaving them and it is not possible to join A to B without crossing another line. So this game was won on the seventh move by the first player.

Play several games with a friend; Sprouts is not as simple as it looks!

Look carefully at some finished games. Regard them as networks. How many arcs are there in each of the networks? How many nodes? What is the order of each of these nodes? Can you explain why a game of sprouts must end after at most 8 moves?

It is possible to vary the game by starting with 2 dots. After how many moves must the game now end?

What happens if you start with 4 dots or 5 dots or ..?

Why is it unsatisfactory to start with only 1 dot?