cube.gms : Three-Dimensional Noughts and Crosses

**Description**

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.

**Reference**

- Williams, H P, Experiments in the formulation of Integer Programming Problems. Mathematical Programming Study 2 (1974).

**Small Model of Type :** MIP

**Category :** GAMS Model library

**Main file :** cube.gms

```
$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).
$Offtext
Sets s domain for line identification / a, b, c, incr, decr /
x(s) coordinate labels / a, b, c /
d(s) directions / incr, decr /
b bounds / low, high /
Alias (x,y,z), (d,dp), (s,sp,spp)
Set ld(s,sp,spp) line definition ;
ld("incr",y,z) = yes; ld(x,"incr",z) = yes; ld(x,y,"incr") = yes;
ld("incr",d,z) = yes; ld(x,"incr",d) = yes; ld(d,y,"incr") = yes;
ld("incr",d,dp) = yes; display ld;
Parameters ls(b) sign for line definitions / low +1, high -1 /
lr(b) rhs for line definitions / low 2, high -1 /
df(x,s) line definition function;
df(x,y) = ord(y) - ord(x); df(x,"decr") = 1 + card(x) - 2*ord(x); display df;
Variables core(x,y,z) placement of balls (white 0 black 1)
line(s,sp,spp) line identification
num number of lines of equal color
Binary Variables core;
Positive Variable line;
Equations nbb total number of balls definition
ldef(s,sp,spp,b) line definitions
ndef number of lines definition ;
nbb.. sum((x,y,z), core(x,y,z)) =e= floor(card(x)**3/2);
ldef(s,sp,spp,b)$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
* this is a very difficult problem that takes a long time with
* the default settings. we use an optca value that will
* cause termination after finding the best solution, but before
* proving that it is the best. the best solution is 4.
Option optca = 3.9;
Solve cube minimizing num using mip;
```