Logic & Mathematics of ColourCross
ColourCross is a Constraint Satisfaction Problem (CSP) based on the principles of Discrete Tomography. While it shares visual similarities with Nonograms (Picross), the underlying logic is fundamentally different.
In traditional Nonograms, the order of the clues matters (e.g., "2 1" implies a block of two, a gap, then a block of one). In ColourCross, clues represent Total Summation. A clue of "2 Red, 1 Blue" simply means there are two red cells and one blue cell somewhere in that line, with no information given about their sequence.
Advanced Solving Techniques
To solve a 5x5 grid without guessing, players must use "Orthogonal Intersection" logic. This involves cross-referencing the constraints of a row against the constraints of its intersecting columns.
1. The "Saturation" Rule
The grid width is exactly 5 cells. If the sum of the clues for a row equals 5 (e.g., 3 Red + 2 Blue), that row is Saturated. It contains no empty spaces. You can immediately fill the row with colours, although you may not yet know which colour goes where.
2. The "Impossible Intersection"
This is the most powerful tool in the Hardest (4-colour) mode.
Example: Row 1 requires a Green cell. However, Column 1's clues show only Red and Blue.
Deduction: It is impossible for R1C1 to be Green. Therefore, the Green cell required by Row 1 must exist in columns 2–5. By marking impossible cells with an 'X', you reduce the permutation space.
3. The "Zero" Constraint
If a row has clues for Red and Blue, but no clue for Yellow, you have established a negative constraint. Every cell in that row is strictly "Not Yellow." When the board becomes crowded, these negative constraints are often the only way to solve the final few cells.
Difficulty Scaling
The complexity of the logic grid increases with the number of variables (colours) introduced:
- Normal (2 Variables): Uses Red and Blue. The logic is binary; if a cell is not empty and not Red, it must be Blue.
- Hard (3 Variables): Adds Yellow. This breaks binary logic, requiring true intersection checking.
- Hardest (4 Variables): Adds Green. With 4 colours and empty space (Void), every cell has 5 potential states. The probability of guessing a correct solution drops to near zero, making strict deductive reasoning mandatory.