\$title Three-dimensional Noughts and Crosses (CUBE,SEQ=42) \$onText White and black balls are to be arranged in a cube one to a cell in such a way as to minimize the number of lines with balls of equal color. For this example the length of the cube is three. a total of 49 lines exist in a cube of 3x3x3. 27 lines by changing only one coordinate, 18 diagonals within a plane, and 4 diagonals going across planes. Williams, H P, Experiments in the formulation of Integer Programming Problems. Mathematical Programming Study 2 (1974). Keywords: mixed integer linear programming, mathematics, mathematical games, noughts and crosses, tic-tac-toe \$offText Set s 'domain for line identification' / a, b, c, incr, decr / x(s) 'coordinate labels' / a, b, c / d(s) 'directions' / incr, decr / b 'bounds' / low, high /; Alias (x,y,z), (d,dp), (s,sp,spp); Set ld(s,sp,spp) 'line definition'; ld("incr",y,z) = yes; ld(x,"incr",z) = yes; ld(x,y,"incr") = yes; ld("incr",d,z) = yes; ld(x,"incr",d) = yes; ld(d,y,"incr") = yes; ld("incr",d,dp) = yes; display ld; Parameter ls(b) 'sign for line definitions' / low 1, high -1 / lr(b) 'rhs for line definitions' / low 2, high -1 / df(x,s) 'line definition function'; df(x,y) = ord(y) - ord(x); df(x,"decr") = 1 + card(x) - 2*ord(x); display df; Variable core(x,y,z) 'placement of balls (white 0 black 1)' line(s,sp,spp) 'line identification' num 'number of lines of equal color'; Binary Variable core; Positive Variable line; Equation nbb 'total number of balls definition' ldef(s,sp,spp,b) 'line definitions' ndef 'number of lines definition'; nbb.. sum((x,y,z), core(x,y,z)) =e= floor(card(x)**3/2); ldef(s,sp,spp,b)\$ld(s,sp,spp).. ls(b)*sum(x, core(x+df(x,s),x+df(x,sp),x+df(x,spp))) =l= line(s,sp,spp) + lr(b); ndef.. num =e= sum((s,sp,spp)\$ld(s,sp,spp), line(s,sp,spp)); Model cube / all /; \$if set nosolve \$exit * note: optCa = 3.9 has been removed due to significant improvements in solver performance solve cube minimizing num using mip;