The square functional equation \[ (1)\quad f(x+t,y+t)+f(x+t,y-t)+f(x-t,y- t)+f(x-t,y+t)=4f(x,y), \] where f: \(R\times R\to R\) and x, y, t are real variables, was investigated by \textit{J. Acźel}, the reviewer, \textit{M. A. McKiernan} and \textit{G.N. Sakovič} [ibid. 1, 37-53 (1968; Zbl 0157.461)]. On the other hand, the functional equation \[ (2)\quad f(x+t,y+t)+f(x-t,y-t)=f(x+t,y-t)+f(x-t,y+t), \] where f: \(R\times R\to R\) and x, y, t are real variables, was investigated by \textit{J. A. Baker} [Can. Math. Bull. 12, 837-846 (1969; Zbl 0187.091)] and the reviewr [Aequationes Math. 5, 118-119 (1970; Zbl 0207.458)]. In this paper the authors study continuous solutions on the real Euclidean n-space, \(R^ n\), of the functional equations \[ (3)\quad \sum^{m}_{j=1}\alpha_ jf(x-ta_ j)=f(x), \] where f: \(R^ n\to R\), \(\alpha_ 1\), \(\alpha_ 2,...,\alpha_ m\in R\), \(\sum^{m}_{j=1}\alpha_ j=1\) and \(a_ 1\), \(a_ 2,...,a_ m\in R^ n\), and \[ (4)\quad \sum^{m}_{j=1}(f(x- ta_ j)-f(x-tb_ j))=0, \] where f: \(R^ n\to R\) and \(a_ 1\), \(a_ 2,...,a_ m\), \(b_ 1\), \(b_ 2,...,b_ m\in R^ n\). The equations (3) and (4) are generalizations of (1) and (2), respectively.