## 6 Numerical Examples
### Tiling Examples Where We Seek a Single Solution  
In this section, we are concerned with finding a single (optimal) solution to the ILP problems associated with tiling. Using checkerboard colouring techniques, we are interested in the pariamino tiling subproblem that yields a feasible solution in the least amount of time. If this is less than the time to find an optimal solution for the full (uncoloured) tiling problem, then we have reduced the complexity of this tiling problem.  
We give a single example to illustrate this approach.  
**Example 13**.: _Find a single way of tiling the 60 \(\times\) 64 rectangle with_  
\[\{P_{1},P_{2}\}=\Bigg{\{}\]  
_all orientations permitted. We aim to tile \(R\) using \(n_{1}\) = 384 copies of the V-shaped pentomino, and \(n_{2}\) = 384 copies of the L-shaped pentomino. The associated ILP file, constructed with the program LPMAKE_POLYONINOES_FIGURE14, has \(n\) = 43,144 unknowns, \(m\) = 3842 constraints, and \(f\) = 39,303 free variables. CPLEX found an optimal solution in approximately 24 h._  
_Using colouring techniques, we wish to tile the 60 \(\times\) 64 checkerboard coloured region \(\widehat{R}\) using 384 pariaminoes from the set_  
\[\{P_{1}^{+},P_{1}^{-}\}=\Bigg{\{}\]  
_with parities \(\pm p_{1}=\pm 1\), and 384 pariaminoes from the set_  
\[\{P_{2}^{+},P_{2}^{-}\}=\Bigg{\{}\]  
Figure 24: Target region and a tiling solution for Example 12: (**a**) the target region \(\widehat{R}\), a checkerboard coloured \(8\times 8\) square; (**b**) successful tiling of the target region \(\widehat{R}\) using five copies of the straight tetromino, seven copies of the \(2\times 2\) square, one copy of the \(2\times 3\) rectangle, and two copies of the P-shaped pentomino. The single optimal solution uses five copies of \(P_{1}^{+}\) (=\(P_{1}^{-}\)), seven copies of \(P_{2}^{+}\) (=\(P_{2}^{-}\)), one copy of \(P_{3}^{+}\) (=\(P_{3}^{-}\)), and one copy each of \(P_{4}^{+}\) and \(P_{4}^{-}\). The pariaminoes \(P_{1}^{+}\), \(P_{2}^{+}\), \(P_{3}^{+}\), \(P_{4}^{+}\), and \(P_{4}^{-}\) are coloured violet, blue, light blue, red, and light red, respectively.  
with parities \(\pm p_{2}=\pm 1\), all orientations permitted. In the interests of space, we do not illustrate \(\widehat{R}\); however, it is sufficient to note that the top left square is white. The parity equation (see Theorem 1) is given by_  
\[p_{1}a_{1}^{+}+p_{2}a_{2}^{+}=(p+p_{1}n_{1}+p_{2}n_{2})/2,\quad 0\leq a_{i}^{+} \leq n_{i},\quad i=1,2.\]  
_After noting that \(p=0\), we obtain the linear Diophantine equation_  
\[a_{1}^{+}+a_{2}^{+}=384,\quad 0\leq a_{1}^{+},a_{2}^{+}\leq 384,\]  
_with 385 solutions \(\{(a_{1}^{+},a_{2}^{+})\ :\ \ a_{1}^{+}=384-a_{2}^{+},\quad a_{2}^{+}=0,1, \ldots,384\}\). Due to time constraints, we only solved a single subproblem corresponding to \((a_{1}^{+},a_{2}^{+})=(0,384)\), i.e., using 384 copies each of \(P_{1}^{-}\) and \(P_{2}^{+}\). We constructed the ILP file in this case using the program LPMAKE_PARIONIONDES_FIGUREE14. Using CPLEY yielded an optimal solution in 2.6 h, with \((m,n,f)=(3842,21572,17732)\). Thus the potential speedup when using the colouring technique is at least 9.2\(\times\). The optimal solution is illustrated in Figure 25._