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For this reason, it is more accurately called "Pólya's problem". The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general, [ 2 ] providing an illustration of the strong law of small numbers .
Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem? Search: Auxiliary Problem: Can you find a subproblem or side problem whose solution will help you solve your problem? Subgoal: Here is a problem related to yours and solved before
The first question sets up an elementary combinatorics question; but the second suggests both a solution (using generating functions) and a generalisation. The third gives another combinatorics question which can be solved by means of generating functions. Indeed, questions 1-26 follow generating function through further examples.
Instead of solving a specific type of problem, which would seem intuitively easier, it can be easier to solve a more general problem, which covers the specifics of the sought-after solution. The inventor's paradox has been used to describe phenomena in mathematics, programming, and logic, as well as other areas that involve critical thinking.
The Polya enumeration theorem translates the recursive structure of rooted ternary trees into a functional equation for the generating function F(t) of rooted ternary trees by number of nodes. This is achieved by "coloring" the three children with rooted ternary trees, weighted by node number, so that the color generating function is given by f ...
Does linear programming admit a strongly polynomial-time algorithm? (This is problem #9 in Smale's list of problems.) How many queries are required for envy-free cake-cutting? What is the algorithmic complexity of the minimum spanning tree problem? Equivalently, what is the decision tree complexity of the MST problem?
One of the reasons for interest in this particular rather elaborate urn model (i.e. with duplication and then replacement of each ball drawn) is that it provides an example in which the count (initially x black and y white) of balls in the urn is not concealed, which is able to approximate the correct updating of subjective probabilities ...
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