Publ. Math. Debrecen 47 / 3-4 (1995), 249–270 On generalized rectangular and rhombic functional equations By J. K. CHUNG (Guangzhou), B. R. EBANKS (Louisville), C. T. NG (Waterloo), P. K. SAHOO (Louisville) and W. B. ZENG (Louisville) 1. Introduction In [2], the functional equations (RE) f (x 1 + y 1 ,x 2 + y 2 )+ f (x 1 + y 1 ,x 2 − y 2 )+ +f (x 1 − y 1 ,x 2 + y 2 )+ f (x 1 − y 1 ,x 2 − y 2 )=4 f (x 1 ,x 2 ) and (RH) f (x 1 + y 1 ,x 2 )+ f (x 1 − y 1 ,x 2 )+ +f (x 1 ,x 2 + y 2 )+ f (x 1 ,x 2 − y 2 )=4 f (x 1 ,x 2 ), among others, were considered for f mapping R 2 into R (the reals). For obvious geometric reasons, these equations are referred to as the rectan- gular and rhombic equations, respectively. Their general solutions are the same: f (x 1 ,x 2 )= A(x 1 ,x 2 )+ B(x 1 )+ C (x 2 )+ α, where A is an arbi- trary biadditive map, B and C are arbitrary additive maps, and α is an arbitrary constant. In the present paper, we generalize those results in three ways. For one thing, we consider a more general right hand side. Also, we generalize the domain to a product of groups and the range to a field. And thirdly, we deal with functions of any finite number of variables. Specifically, we consider the equations (GRE)  σ 1 ,...,σ n =±1 f (x 1 y σ 1 1 , ..., x n y σ n n )= = f (x 1 , ..., x n ) g(y 1 , ..., y n )+ h(y 1 , ..., y n )