SOLVING THE KAKURO PUZZLE
Andreea Erciulescu
Department of Mathematics,
Colorado State University, Fort Collins
(Mentor: A. Hulpke)
Abstract
Kakuro puzzles are NP-complete ("Non-deterministic
Polynomial time"). Although brute-force guessing is a possible
way to solve them, a better weapon is the understanding of
the various combinatorial forms that entries can take for
various pairings of clues and entry lengths. In this
presentation I will introduce you to one approach towards
completing an algorithm to solve Kakuro without guessing.
Introduction
Puzzle's Definition
• is a kind of logic puzzle that is often referred to as a
mathematical transliteration of the crossword
• is given on a grid of black and white cells
• the digits 1 to 9 must be filled into all the white cells so that they
satisfy the clues given in some of the black cells
• the clues specify the sum of the numbers in the row of successive
white cells to the right or the column of successive white cells below
• no row or column of successive white cells can have a digit
repeated
These puzzles are regular features in most, if not all, math-and
logic puzzle publications in the United States. The popularity of Kakuro
in Japan is immense, second only to Sudoku.
You can try out some of these puzzles on the following website:
http://www.kakuropuzzle.com
A 4 x 4 example
Methods
• We consider this puzzle a linear problem with variable entries and
we solve it over Z
• Smith Normal Form and Lattice Reduction
• Combinatorial search
Compute the integer solutions
Let the matrix of equations be N, defined as follows:
N is a n_row x n_col matrix with entries in {0,1}, where n_row
represents the number of equations and n_col represents the number
of unknown variables.
Compute the integer solutions
Smith Normal Form
• X = the vector of unknown variables (the solution) in the range 1..9
• B = the matrix whose entries are the clues from the black cells, in
the puzzle
B=(4,7,6,4,7,6)T
• Need to solve the system Nx=B over the Z-module.
• By Smith Normal Form, there are invertible matrices P and Q over
the Z-module such that PNQ=M, where M is in Smith Normal Form.
• From the system x=QY and MY=P-1B we have the solution to the
inhomogeneous equation and the solution to the homogeneous
equation MY= 0.
• However, the entries are possibly large. For this, we implement
Lattice Reduction to find the small integer solutions.
Find the small integer solutions
• We use Lattice Reduction on the set formed with both the
homogeneous and the inhomogeneous solutions and as solution
get the rows of
and the resulting set of vectors spans the same initial space.
Find the small integer solutions
• Now we have the homogeneous solutions and the partial one,
which satisfies the linear system, given by the rows of the matrix
• Since most of the times it is not a solution for the puzzle, we need
to proceed to the next step.
Combinatorial Search
• Consider all combinations of partial + ∑icihi , where hi are the
homogeneous solutions and ci are small integers that would give
small entries in the solution
• Look only for the solutions that satisfy the puzzle ( numbers
between 1 and 9 and no duplicates on the same column and same
row)
• A possible solution is a linear combination of the reduced basis
vectors
• Use backtracking for examining feasible solutions, by systematically
eliminating infeasible solutions
Combinatorial Search
• We first detect the vectors hi that have few nonzero entries
corresponding to the entries in the particular solution that need
to be changed. For these, we test the coefficients ci and prune
the tree whenever the choice determines the entry in the
solution.
• Observe that lll[4]-lll[3]-lll[2] = [1,3,1,4,2,3,2,1] is the
solution to the puzzle.
Future Directions and Conclusions
• Solving a Kakuro puzzle is a NP problem. The complexity class NPcomplete (standing for Nondeterministic Polynomial time) is often found
in the computational complexity theory. For these problems, one can
quickly verify a solution, whenever it is given. However, there is no
efficient way to think about a solution in the first place. Solving these
problems requires computer algorithms and a large amount of time.
• We have tested puzzles up to the size 25x35 and the algorithm gives the
right solution.
• We are looking at a better way to enter the information from the puzzle.
• We would like to eliminate the infeasible solutions earlier.
• We would like to know what would be the exact range of coefficients that
gives the solution.
Bibliography
• Morandi, Patrick, The Smith Normal Form of a Matrix , 17 February,
2005
• Lenstra, Lenstra, Lovasz L., Factoring Polynomials with Rational
Coefficients,1982
Thank you!