### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

### Pattern of Islands

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

### Only Connect

The graph represents a salesman’s area of activity with the shops that the salesman must visit each day. What route around the shops has the minimum total distance?

# Konigsberg Plus

##### Stage: 3 Challenge Level:

We received three good solutions - two almost completely correct, fromYanqing from Lipson Community College and from Sarah from Colyton, and one completely correct from Andrei from Tudor Vianu College in Bucharest. Well done to you all.

Andrei's solution follows:

First, I observe that for the first figure it is impossible to cross each bridge once. This happens because you would remain in a place where all the bridges were already crossed.

In the second case, it is possible to cross the bridges only once and create a circuit.

I worked with different number of bridges and islands and I obtained solutions for each of the following:
- bridge traversing circuits
- bridge traversing paths
- Hamiltonian circuits
- Situations where none of the above is possible.

For 8 bridges I found:

- A diagram with a bridge traversing circuit.

The circuit is the following: S-C-A-D-B-A-C-B-S

- A diagram with a bridge traversing path:

The path is the following: S-D-E-A-B-S-C-E-B.

- A Hamiltonian circuit and a bridge traversing circuit:

- A diagram in which neither is possible. In the diagram below, neither a traversing path nor a traversing circuit is possible:

Editor's note:
If you were allowed to start at another island you could have a traversable path: starting at the top right hand island and finishing in the top left hand island, or vice versa.