SOME PROPERTIES OF MAGIC CUBES

Abstract

Abstract. In this paper we introduce an algorithm to construct an in- nite family of magic cubes and investigate properties of magic cubes,which extend the results in [6]. 1. IntroductionThe study of magic squares has a long history([4]). Some remarkable prop-erties of magic squares have been shown and some generalizations have drawnon interesting mathematics([1] and [3]). In [2] an operation on the set of allmagic squares was introduced which makes the set of magic squares into a freemonoid. In [6] a product on the set of all matrices was introduced and an al-gorithm was given to construct an in nite family of magic squares. Under theproduct operation, it was shown that the set of all magic squares formed a freemonoid as in [2] and moreover the product preserved the symmetrical propertyintroduced in [5] to give in nitely many square-palindromic magic squares. Inthis article, we extend the results in [6] to construct an in nite family of magiccubes and investigate properties of magic cubes.2. Cube algebraAn n-cube is an n-dimensional array of real numbers. When n= 2;we haveusual matrices. For an n-cube A;every component can be written by an n-aryexpression a