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Statement

 Goal

In mathematics and computability theory, an elementary cellular automaton is a one-dimensional cellular automaton where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors.

The numbering system

There are 2^3=8 possible configurations for a cell and its two immediate neighbors. The rule defining the cellular automaton must specify the resulting state for each of these possibilities so there are 2^(2^3) = 256 possible elementary cellular automates. Stephen Wolfram proposed a scheme, known as the Wolfram code, to assign each rule a number from 0 to 255 which has become standard. Each possible current configuration is written in order, 111, 110, ..., 001, 000, and the resulting state for each of these configurations is written in the same order and interpreted as the binary representation of an integer. This number is taken to be the rule number of the automaton. For example, 110 (base 10)=01101110 (base 2). So rule 110 (ruleNumber) is defined by the transition rule:

=================================
|111|110|101|100|011|010|001|000|
| 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |
=================================


You can find similar a puzzle in coding game here: https://www.codingame.com/training/medium/elementary-cellular-automaton

More information about elementary cellular automaton can be found here : http://mathworld.wolfram.com/ElementaryCellularAutomaton.html

Statement

based on lineLength an odd number, start with a line with 1 in the middle and fill all the rest with '.' char.

'.' character represent the 0 value, it is used to make the display more readable.
'1' character represent the 1 value.

Example:
line_length = 1 : 1 for 1
line_length = 3 : .1. for 010
line_length = 5 : ..1.. for 00100

For this exercise, we connect the two extremities of the line, think about a motif that will be displayed over an infinite cylinder

The left cell of the first cell is the last cell.
The right cell of the last cell is the first cell.

Then repeat the transformation to the current line till you find a repetition scheme, a line that was already outputted before.

Since same line will always provide same output, we can be sure that a repetition period exist and this period is less or equal to 2^(lineLength) + 1

Goal

Output the repetition period (the number of iteration separating two matching patterns), if the number of iteration comes greater than maxIter and no solution was found yet, output "BIG".

Example 1:

- lineLength = 5
- maxIter = 20000
- ruleNumber = 254

The lines will be like that:

..1..
.111.
11111
11111


After the second line, the lines stay the same forever the the repetition period is then 1

Example 2:

- lineLength = 5
- maxIter = 20000
- ruleNumber = 1

The lines will be like that:

..1..
1...1
..1..


Period is 2

bonus

Display each line of iteration inside the error stream to better understand how the chaos can emerge from a simple defined problem.
Input
Line 1: lineLength the length of the line to repeat
Line 2: maxIter maximum number of iteration
Line 3: ruleNumber the rule number
Output
The repetition period or "BIG" if no period can be found after maxIteriterations
Constraints
lineLength is odd and < 500
maxIter < 50000
0 ≤ ruleNumber ≤ 255
Example
Input
5
20000
254
Output
1

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