Calculating the Odds of Winning in the Well Game: A Comprehensive Guide

[ryan8200]

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?

 

[dikmikkel]

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.

 

[ryan8200]

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.

 

[dikmikkel]

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.

 

[jing2178]

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.