The Kakuro puzzle is another logical puzzle similar to Sudoku, KenKen and Takuzu. A small example from (1) is:
The rules are as follows:
- Each (white) cell must be filled with a value \(1,..,9\).
- Each (white) cell belongs to a horizontal and a vertical block of contiguous cells.
- The values of the cells in each block add up to a given constant. These constants are to the left (or top) of the block. A notation a\b is used: a is the sum for the block below and b is the sum for the block to right.
- The values in each block must be unique (i.e. no duplicates).
Model
Similar to (2) and (3) I define a binary decision variable:
| \[x_{i,j,k} = \begin{cases}1 & \text{if cell $(i,j)$ has the value $k$} \\ 0& \text{otherwise}\end{cases}\] |
where \(k=\{1,..,9\}\). Of course, we can select only one value in a cell, i.e.:
| \[ \sum_k x_{i,j,k} = 1 \>\>\forall i,j|IJ(i,j)\] |
where \(IJ(i,j)\) is the set of white cells we need to fill in. To make the model simpler we also define an integer variable \(v_{i,j}\) with the actual value:
| \[v_{i,j} = \sum_k k\cdot x_{i,j,k}\] |
The values in each block need to add up the given constant. So:
| \[\sum_{i,j|B(b,i,j)} v_{i,j} = C_b \>\>\forall b\] |
where the set \(B(b,i,j)\) is the mapping between block number and cells \((i,j)\). The uniqueness constraint is not very difficult at all:
| \[\sum_{i,j|B(b,i,j)} x_{i,j,k} \le 1 \>\>\forall b,k\] |
Data entry
A compact way to enter the data for the example problem in GAMS is:
parameter |
The sets \(IJ(i,j)\) and parameter \(C_b\) can be easily extracted from this.
Results
After adding a dummy objective, we can solve the model as a MIP (Mixed Integer Programming) Model. The output looks like:
---- 75 VARIABLE v.L c1 c2 c3 c4 c5 c6 c7 r1 9 7 8 7 9 |
It is noted that although I used the declaration \(x_{i,j}\), only the subset of cells that we actually need to fill in is used in the equations. GAMS will only actually generate the variables that appear in the equations, so all the “black” cells are not part of the model.
Cplex can solve the model in zero iterations and zero nodes: it solves the model completely in the presolve.
Interestingly in (1) somewhat disappointing performance results are reported. They use CBC and a solver called eplex which I had not heard about before. I checked the above small instance with CBC and it can solve it in zero nodes, just like Cplex.
References
- Helmut Simonis, Kakuro as a Constraint Problem, Cork Constraint Computation Centre (4C), University College Cork, 2008.
- MIP Modeling: From Sukodu to Kenken via Logarithms, http://yetanothermathprogrammingconsultant.blogspot.com/2016/10/mip-modeling-from-sudoku-to-kenken.html
- Solving Takuzu puzzles as a MIP using xor, http://yetanothermathprogrammingconsultant.blogspot.com/2017/01/solving-takuzu-puzzles-as-mip-using-xor.html