Suppose a biking environment consists of n ≥ 3 landmarks, which are linked by bike route in a cyclical manner. That is, there is a bike route between landmark 1 and 2, between landmark 2 and 3, and so on until we link landmark n back to landmark 1. In the center of these is a mountain which has a bike route to every single landmark. Besides these, there are no other bike routes in the biking environment. You can think of the landmarks and the single mountain as nodes, and the bike routes as edges, which altogether form a graph G. A path is a sequence of bike routes. What is the number of paths of length 2 in the graph in terms of n? What is the number of cycles of length 3 in the graph in terms of n? What is the number of cycles in the graph in terms of n?

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Heuristic System

The term heuristic is derived from the Greek word heurisko, which means “I find, discover”. Heuristics can be mental shortcuts or approaches designed from previous experiences. The main aim of the heuristic approach is to find a good enough solution to solve the problem. Test and mistake, rule of thumb, or an educated estimate are all examples of heuristics. These methods are sufficient for achieving an immediate, brief aim. When a heuristic is used in practice, it works as predicted. It can, however, produce systematic errors. Trial and error is an important heuristic that may be used in a variety of situations, from cross-linking nuts and bolts to determining the values of given input variables in mathematical puzzles. Additional hypotheses, visual illustrations, simplification, and forward or backward reasoning are all common heuristics in mathematics.