How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.

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

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.

A problem that did not need high level mathematics just clear thinking and a little trigonometry. Well done to the large number of you who achieved a correct solution, including Barinder, Roy, Dan and Calum - to name a few.

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.