# Problem Solving

## CONTENTS OF CURRICULUM UNIT 80.07.08

## Topology

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## 3. ARCS AND REGIONS

Each of the networks in figure 25 has 5 nodes, 8 arcs and 5 regions. (DO NOT COUNT 2NODES.)

### INVESTIGATION 2

SUGGESTION: You can ask students to make an addition column for N+R.

- (a) Complete the table showing the number of nodes (N), arcs (A) and regions (R) for the networks in figure 26. NOTE: Provide each student a copy of figure 26 and add 7 more networks.
- (b) Ask your neighbor to check your results.
- (c) Look for patterns in your table. Comment on these patterns. NOTE: It is possible that the result will be found in many forms, such as N + R = A2, NA + R = 2. This can be followed by a useful discussion as to whether or not these formulas are equivalent. Other results which have been noted by students include: (i) N + A + R is always even, (ii) A + RN is always even. It should be noted that the formula N + R = A+2 remains true when a finite number of 2 nodes are counted. In general, 2nodes may be ignored.