Odds of getting 1, 2, ... 8 lines in the well game

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https://www.physicsforums.com/attachments/46168

This is the "well" game (in Chinese) similar to Sudoku. It has 9 areas with different colours. I will draw one number from each area. In order to have a line, at least 3 number drawn must be same row, column or diagonal. I will illustrate an example as below:
https://www.physicsforums.com/attachments/46169

How to calculate the odds of having 1,2...8 lines?

Anyone can help me or give me some hints?
 
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I use this as a general rule, but i don't really get the game:
Every time i say and i multiply: What is the probability of a 6-sided dice showing 1 twice(and 1) in a row: 1/6*1/6 = 1/36
Every time i say or, i add so: What is the probability of a 6-sided dice showing 1 twice in a row or 3 - three times in a row: 1/6*1/6 + 1/6*1/6*1/6 = 1/36 + 1/216 = 6/216+1/216 = 7/216.
Hope it helps a little.
 
dikmikkel said:
I use this as a general rule, but i don't really get the game:
Every time i say and i multiply: What is the probability of a 6-sided dice showing 1 twice(and 1) in a row: 1/6*1/6 = 1/36
Every time i say or, i add so: What is the probability of a 6-sided dice showing 1 twice in a row or 3 - three times in a row: 1/6*1/6 + 1/6*1/6*1/6 = 1/36 + 1/216 = 6/216+1/216 = 7/216.
Hope it helps a little.

The mechanism of the game is illustrated as below,I hope it will help. My main concern is how to calculate the total line formed for all possible numbers been drawn.

Capture3.jpg
 
You could maybe make a matrix describing the game and make a program that plays the game a hell of times and calculate the probabilities so make the program regognize the type of win diagonal etc.
See it as a nummerical approach maybe.
 
As a start think about this.
How do you get a line along the top row?
What is the probability of choosing a number in the first column top row, second column top row, third column top row?

If you have a line in the top row is it possible to choose numbers elsewhere so that no more lines can be drawn?
If you have a line in the top row is it possible to choose numbers elsewhere so that 1 more line can be drawn?
If you have a line in the top row is it possible to choose numbers elsewhere so that 2 more lines can be drawn?
etc.
 
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