where k B is the Boltzmann constant, N is the number of oxygen atoms in the piece of ice, which is always taken to be large (the thermodynamic limit) and Z = W N is the number of configurations of the hydrogen atoms according to Pauling's ice rule. Without the ice rule we would have W = 4 since the number of hydrogen atoms is 2 N and each hydrogen has two possible locations. Pauling estimated that the ice rule reduces this to W = 1.5 , a number that would agree extremely well with the Giauque-Stout measurement of S . It can be said that Pauling's calculation of S for ice is one of the simplest, yet most accurate applications of statistical mechanics to real substances ever made. The question that remained was whether, given the model, Pauling's calculation of W , which was very approximate, would be sustained by a rigorous calculation. This became a significant problem in combinatorics.