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Counting the number of N-step self-avoiding walks (SAWs) on a lattice is oneof the most difficult problems of enumerative combinatorics. Once we give upcalculating the exact number of them, however, we have a chance to applypowerful computational methods of statistical mechanics to this problem. Inthis paper, we develop a statistical enumeration method for SAWs using themulticanonical Monte Carlo method. A key part of this method is to expand theconfiguration space of SAWs to random walks, the exact number of which isknown. Using this method, we estimate a number of N-step SAWs on a squarelattice, c_N, up to N=256. The value of c_256 is 5.6(1)*10^108 (the number inthe parentheses is the statistical error of the last digit) and this is largerthan one googol (10^100).

How to estimate the number of self-avoiding walks over 10^100? Use random walks