This paper presents efficient hypercube algorithms for solving triangular systems of linear equations by using various matrix partitioning and mapping schemes. Recently, several parallel algorithms have been developed for this problem. In these algorithms, the triangular solver is treated as the second stage of Gauss elimination. Thus, the triangular matrix is distributed by columns (or rows) in a wrap fashion since it is likely that the matrix is distributed this way after an LU decomposition has been done on the matrix. However, the efficiency of the algorithms is low. Our motivation is to develop various data partitioning and mapping schemes for hypercube algorithms by treating the triangular solver as an independent problem. Theoretically, the computation time of our best algorithm is (\left((12 p+1) n^{2}+36 p^{3}-\right.) (\left.28 p^{2}\right) /\left(24 p^{2}\right)), and an upper bound on the communication time is (2 \alpha p \log p(\log n-\log p)+2 \alpha(\log n-\log p-1) \log p+(c n / p-) 2c) ((2 \log p-1)+\log p(c n-c-\alpha)), where (\alpha) is the (communication startup time)/(one entry scanning time), (c) is a constant, (n) is the order of the triangular system and (p) is the number of nodes in the hypercube. Experimental results show that the algorithm is efficient. The efficiency of the algorithm is 0.945 when (p=2, n=) 513, and 0.93 when (p=8, n=1025). 1994 Academic Press. Inc.