funsters, over the last month, i've become interested in the following type of chess problem: given two types of chess pieces (possibly equal, no pawns) A and B, and two positive integers m and n, is it possible to find a placement of A's and B's so that each A attacks exactly n B's (and no A's), and each B attacks exactly m A's (and no B's)? if so, what is the smallest rectangular board (in terms of area) on which this is possible? i have found many such arrangements, which can be found here: http://www.stetson.edu/~efriedma/mathmagic/0207.html does anyone know if this problem has been considered before? i am convinced that other cases are possible, and that many of my solutions can be improved. can anyone solve any of the unsolved cases, or prove they are impossible? can anyone find smaller solutions for some of these cases? other variations on this theme of pieces attacking only one other type of piece are also explored on the above page. my favorite, and one of the hardest, is: given three types of chess pieces A, B, and C, is it possible to find a placement of A's, B's and C's so that each A attacks 2 B's, each B attacks 2 C's, and each C attacks 2 A's (with no other attacks)? if so, what is the smallest solution? the solutions with A=bishop, B=king, C=knight and A=queen, B=king, C=knight were especially fun to find. erich friedman