Assume that multiplying a matrix  of dimension  with another matrix  of dimension  requires scalar multiplications. Computing the product of  matrices  can be done by parenthesizing in different ways. Define  as an explicitly computed pair for a given parathesization if they are directly multiplied. For example, in the matrix multiplication chain  using parenthesization  and  are the only explicitly computed pairs.
Consider a matrix multiplication chain  where matrices  and  are of dimensions   and  respectively. In the parenthesization of  that minimizes the total number of scalar multiplications, the explicitly computed pairs is/are