Random Numbers. How are the various decisions made? To start with, the computer must have a source of uniformly distributed psuedo-random numbers. A much used algorithm for generating such numbers is the so-called von Neumann middle-square digits. Here, an arbitrary n-digit integer is squared, creating a 2n-digit product. A new integer is formed by extracting the middle n- digits from the product. This process is iterated over and over, forming a chain of integers whose properties have been extensively studied. Clearly this chain of numbers repeats after some point. H. Lehmer has suggested a scheme based on the Kronecker-Weyl theorem that generates all possible numbers of n digits before it repeats. (See Random-Number Generators for a discussion of various approaches to the generation of random numbers.)

Once one has an algorithm for generating a uniformly distributed set of random numbers, these numbers must be to transformed into the nonuniform distribution g desired for the property of interest. It can be shown that the function f needed to achieve this transformation is just the inverse of the nonuniform distribution function, that is f=g ^-1 . Nicholas C. Metropolis