An improved lower bound for the multidimensional dimer problem

The dimer problem, which in the three-dimensional case is one of the classical unsolved problems of solid-state chemistry, can be formulated mathematically as follows. We define a brick to be a d-dimensional (d ≥ 2) rectangular parallelopiped with sides whose lengths are integers. An n-brick is a brick whose volume is n; and a dimer is a 2-brick. The problem is to determine the number of ways of dissecting an n-brick into dimers; and since this is only possible when n is even we confine attention hereafter to n-bricks with n even. Consider an n-brick with sides of length a1, a2, …, ad, where n = a1a2 … ad, and write a = (a1, a2, …, ad). Let fa denote the number of ways of dissecting this brick into ½n dimers. On the basis of physical and heuristic arguments chemists have known for many years that fa increases more or less exponentially with n; and recently a rigorous proof (1) of this fact has been given in the following form: if ai → ∞ for all i = 1, 2, …, d, then n−1 logfa tends to a finite limit, which we denote by λd. The principal outstanding problem for chemists is to determine the numerical value of λ3, or failing an exact determination to estimate λ3 or to find upper and lower bounds for it.

On: “THE GRAVITATIONAL ATTRACTION OF A RIGHT RECTANGULAR PRISM,” BY DEZSÖ NAGY (GEOPHYSICS, APRIL, 1966, PP. 362–371)

You might be interested to note that the results obtained by Nagy (1966) and published in Geophysics concerning the vertical component of the gravitational attraction of a right rectangular prism not only had been published previously by Sorokin and Haáz but also were derived and published (in English) 136 years ago by Everest (1830, p. 94–97). Everest calculated closed expressions which are equivalent to that of Nagy for the horizontal and vertical gravitational effects of a rectangular parallelopiped and used these equations to estimate the topographic deflection of the plumb bob due to the Satpura Range in India.

IV. On an approximate solution for the bending of a beam of rectangular cross-section under any system of Lond, with special reference to points of concentrated or discontinuous loading

The consideration of the stresses and strains which occur in a rectangular parallelopiped of elastic material subjected to given surface forces over its six faces leads to one of the most general, as it is one of the oldest, problems in the Theory of Elasticity. Lame, in his ‘Lemons sur l’Elastieite des Corps solides,’ published in 1852, describes it as “le plus difficile peut-etre de la theorie mathematique de ’elasticity.” In spite of repeated attempts, however, the problem remains still unsolved. In its complete form it may he stated as follows:—