# Math

A mathematical model of the problem.

## Variables

• A boolean variable $\text{cell}_{i,j}$ per cells of the grid, set to true when the cell is shaded, false oterhwise.

$$\forall i,j \in [1,15]^2, \text{cell}_{i,j} = \{0,1\}$$

And that’s enough !

## Constraints

Any column and row should respect its sequence in BLOCKS. To do so, each sequence of BLOCKS is turned into a Deterministic Finite Automaton (DFA). There is a gap between two consecutive blocks, before the first block and after the last block of a sequence there can be a gap. The length of each block is injected in the DFA. Any gap is encoded with a ‘0’s and any block by ‘1’s.

The constraint needed here is named regular. It takes a finite sequence variables and a DFA as input. With the help of a vocabulary (available values, in our case 0 and 1) and a grammar (the allowed sequence), the constraint constructs only valid words of a given size (the variables length). It ensures a very good level of filtering (AC), but the major difficulty relies on building the DFA.

There are two ways to create it: based on a regular expression or describing all states and transitions.

## Objective

The objective is to find a solution that satisfies each constraint.