Networks/graph theory

problem
Pattern of islands

Age

11 to 14

Challenge level

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

Age

11 to 14

Challenge level

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
problem
Cube net

Age

16 to 18

Challenge level

How many tours visit each vertex of a cube once and only once? How many return to the starting point?
problem
Instant insanity

Age

11 to 18

Challenge level

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.
problem
Magic caterpillars

Age

14 to 18

Challenge level

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.
problem
Fermat's poser

Age

14 to 16

Challenge level

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
problem
Limiting probabilities

Age

16 to 18

Challenge level

Given probabilities of taking paths in a graph from each node, use matrix multiplication to find the probability of going from one vertex to another in 2 stages, or 3, or 4 or even 100.
problem
Networks and nodes

Age

7 to 11

Challenge level

Without taking your pencil off the paper or going over a line or passing through one of the points twice, can you follow each of the networks?
problem
Redblue

Age

7 to 11

Challenge level

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
problem
Hamilton's puzzle

Age

7 to 11

Challenge level

I start my journey in Rio de Janeiro and visit all the cities as Hamilton described, passing through Canberra before Madrid, and then returning to Rio. What route could I have taken?