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\noindent
Problem-Solving Principles

\vskip .125in

When a programming problem seems too hard to make progress on, here are some
standard paradigms $\lbrack$ Polya $\rbrack$ for getting started:

\ip{(1)}
Is there a special case which is easier than the general problem, for which a
solution would help with the general problem?

Example: I want a formula for the area of a polygon, given the coordinates of
its vertices. If I can solve the problem for triangles, I may be able to cut
all other polygons into triangles.

\ip{(2)}
Is there some more general problem I should be looking at, with my original
problem as a special case?

Example: I am trying to design an algorithm to compute $A↑B$, for $O≤B$ an
integer. A more general problem is to compute $A↑B\times C$. 
By setting $C=1$, an algorithm for the general problem solves the more
specialized one. The more general problem suggests the operations

$B:=B-1;\; C:=C*A.$

\noindent
and (when B is even)

$B:=B\;DIV\; 2;\; A:=A*A.$

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