Unsolved Problems in Computational Science III: A Special Self-Avoiding Walk

In our "Unsolved Problems in Computational Science", we will talk about some interesting open problems in computational science. These problems are related to number theory, geometry theory, combinatorics, graph theory, linear algebra and group theory. They are easy to state and understand although they are very difficult to be solved by researchers in mathematics or computer science. It seems to us that it is very challenging to find suitable mathematical methods or efficient algorithms to deal with them. In this paper, we will present: A self-avoiding walk (SAW) is a sequence of moves on a lattice not visiting the same point more than once. A SAW on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most sub-classes of SAWs that have been counted so far in that they can wind around their starting point. Some problems and some sequences arising from prudent walks are discussed in this paper.