Networks are the theoretical foundations of many models that we use in everyday life. Examples are street maps or the map of the subway in every major city. The aim is usually to keep these networks clear by visualizing them well and keeping structures easily manageable.

A network can quickly become confusing. All you need to do is add a few nodes and several connections.

The Hirnsport.de team has developed new brain teasers based on this approach. How well can you orientate yourself in a network or do you quickly lose sight of the goal. Try it!

## The task: Orientate yourself in the network

Find a path through the network from start to finish where the sum of the visited nodes meets certain conditions. Each rectangular node may only be visited exactly once. Possible connections between the nodes are shown as direct lines. The direction of the connection is not specified and both ways are possible.

Example:

A plan for a network is given. Find a path from start to end and sum the numbers of nodes on the route. Each node may only be visited exactly once. The visible connections between the nodes can be used in both directions.

A sample path:

Start ▷ 10 ▷ 8 ▷ -3 ▷ End. The path sums up to the total of 15

## The Exercises

**Brain Teaser 1**

Find the path with the highest sum of visited knots.

Find the path with the lowest sum of visited knots.

## Solution for brain teaser 1

The path with the lowest total:

Start ▷ 6 ▷ -4 ▷ 3 ▷ -2 ▷ -5 ▷ -3 ▷ End. The path sums up to the total of -5.

The path with the highest total:

Start ▷ 6 ▷ -4 ▷ 10 ▷ 8 ▷ 3 ▷ -2 ▷ End. The path sums up to the total of 21.

**Brain Teaser 2**

Find the path with the highest sum of visited knots.

Find the path with the lowest sum of visited knots.

## Solution for Brain Teaser 2

The path with the lowest total:

Start ▷ -1 ▷ -3 ▷ -2 ▷ -4 ▷ 2 ▷ End. The total of this path is -8.

The path with the highest total:

Start ▷ 10 ▷ 1 ▷ 2 ▷ -4 ▷ 9 ▷ -3 ▷ 6 ▷ End. The total of this path is 21.

**Brain Teaser 3**

Find the path with the highest sum of visited knots.

Find the path with the lowest sum of visited knots.

Additional exercise: Find the path with a total of exactly zero.

## Solution for brain teaser 3

The path with the lowest total:

Start ▷ -1 ▷ -7 ▷ -4 ▷ 7 ▷ -8 ▷ End. The total of this path is -13.

The path with the highest total:

Start ▷ 11 ▷ 19 ▷ 18 ▷ 7 ▷ -4 ▷ 12 ▷ 17 ▷ End. The total of this path is 80.

The path where all knots sum up to the total of zero:

Start ▷ -4 ▷ 12 ▷ -8 ▷ End. The total of this path is 0.