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).
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;
```