A NALYSIS OF THE FRENCH ONE - PLAYER GAME K AMAJI
K ELVIN K LEIJN
L EIDEN I NSTITUTE
AND J EANETTE DE
OF
G RAAF
A DVANCED C OMPUTER S CIENCE
[email protected]
I NTRO
R ESEARCH Q UESTION
The game of K AMAJI is a game that originated in France and is played by one player.
The gameplay consists of an N × N twodimensional grid filled with N × N squares
that all contain numbers, just as in a Sudoku puzzle. One of the squares is coloured
blue and contains the maximum value. All
the other squares contain numbers that are
strictly smaller than this value. The goal is to
draw horizontal, vertical and diagonal lines
that cover the squares, such that each square
is covered by only one line, except the ones
containing a 1. Those squares can be covered
any number of times. In addition, the numbers contained by the squares covered by a
line must add up to the maximum value. So,
the blue-coloured square is not covered by any
line. Below is an example of a 4×4 Kamaji. The
second figure is a solution to the Kamaji puzzle shown in the first figure.
Research will be conducted into several aspects of Kamaji puzzles, including the following:
• The complexity of Kamaji puzzles. Particularly, what makes a Kamaji puzzle relatively difficult or easy? What factors are relevant to this question?
• What different approaches are there to finding a solution to Kamaji puzzles? Which of those
are effective in terms of complexity and how do they compare to other strategies? There are
common algorithms that can be considered such as backtracking, but also strategies that
are more specific to Kamaji puzzles such as developing a solution by expanding a potential
solution from the corners to the middle of the grid.
• Once we have identified several factors that affect the difficulty of a Kamaji puzzle, can we
generate a uniquely-solvable Kamaji puzzle of a given difficulty?
S OLUTION S TRATEGIES
General Backtracking
Backtracking can be used to solve Kamaji puzzles. We apply Backtracking to find a solution by
beginning the search, starting from the upper-left square. For each square we try to expand in
each direction, searching for combinations that add up to the maximum value. When a square
has been covered, we continue the search for the solution by continuing with the next entry in
the row. When all entries within a row have been covered we continue with the first entry in the
next row. This algorithm allows us to ignore the left, upper-left, top and top-right neighbours
when we try to expand from a certain square, because the order ensures that these squares have
been covered already.
Rules of Thumb
• First, combine all squares that contain (the maximum value - 1) and have a unique adjacent
1. All these must be surrounded by at least a single 1.
• Expand the possible solution from the corners to the middle of the grid. The lower the
number of adjacent neighbours that can be expanded, the more likely that there is only one
valid combination.
• Trying to make combinations starting from the larger values first. Generally speaking,
higher values lead to combinations involving fewer squares.
R ELATED W ORK
Intro
In certain cases, distinct valid combinations
lead to multiple solutions. For instance, as we
can see in the above example, the square containing the number 4 that is combined with
the bottom-left one, also could had been combined with the square above it. Multiple solutions may exist even if no 1’s are involved.
Unfortunately, not much information on Kamaji puzzles is available. There are other puzzles
that involve two-dimensional grids and numbers. Two prominent examples are Sudoku’s and
Kakuro’s. These have been proved to be NP- complete. Although these may seem to be good candidates for a comparison, upon closer examination it seems that Word Search puzzles are even
more similar to Kamaji puzzles, especially in terms of complexity, despite the fact that these puzzles involve words instead of numbers. Therefore, the next section describes a basic comparison
between The Kamaji Puzzle and the Word Search puzzle.
The Word Search Puzzle And The Kamaji Puzzle: A Comparison.
There are obvious similarities between Kamaji puzzles and Word Search puzzles.
• Both involve a two-dimensional grid.
Kamaji in Education
• Squares must be combined to form groups.
The Freudenthal Institute Encourages the utilization of Kamaji puzzles within the education sphere to:
• Squares can be combined horizontally, vertically and diagonally.
• Enhance problem solving ability.
• Promote cooperation and differentiation.
• Improve basic math.
http://www.fi.uu.nl/
Nevertheless, there are also some differences between the two:
• First, In Kamaji puzzles, all entries must be covered. This is not true for Word Search puzzles.
• Second, In Word Search puzzles, all entries may be used any number of times.
• Third, every combination that can be found in Word Search puzzles is valid. In Kamaji’s,
this is only true if it can lead to a solution.