WebSudoku at the Evil Level: 10 plus Billion Unique Games? I Kinda Doubt It

I have recorded game information from 1000 plus Sudoku games to first study random number generation and then secondly, began looking at the level of uniqueness in Sudoku game patterns that are offered by WebSudoku dot com.

This has been a fun mathematics project that has evolved over time with the addition of new techniques & discoveries.

After recording several games at the Evil level, there appear to be about ten and half billion games that could possibly show up based on web page refresh or actually selecting the button to get a new game. 

After collecting data looking at the spread of game numbers that are selected randomly by the system, I decided to begin recording the game locations using an octal numbering system for recording which game cells are filled at the start of the game, 'Given Cells'. I have recorded game information about Given Cells in an Excel spreadsheet in order to sort the information in a variety of ways to look for repeated patterns. I wasn't recording the exact values in cells, merely the pattern of Given Cells in the puzzle patterns. After finding repeated Givens patterns, I followed up with actual game play of the repeated Givens to prove that multiple games are merely logical repeats of full game play.

In the Excel spreadsheet there are two columns for Givens pattern with Octal values, with this game having  000,261,744,604,050 for rows 5 through 9 and a calculated subset, of rows 5 and 6: 604050 for easier sorting & searching. The column display format has commas for legibility 

WebSudoku games at the Evil level appear to be symmetrical so that each half are the same with rotational symmetry in regards to the placement of Given Cells. So each cell that is exactly opposite in the game pattern shares the same logic of either being a Given Cell, or empty. So recording the information about the top 5 rows of a puzzle is more than enough to describe the pattern of Given Cells, but not the unique value (1-9) of the cells. Rotating the entire game 180 degrees gives the same pattern of Givens but not necessarily the same unique cell values for the opposite cells.

Two Sudoku games are essentially the same game if all the numbers of Game A can be translated one-to-one with Game B.

Sudoku Game A can also be compared to another Game B after transforming it by flipping horizontally or vertically and/or rotating the entire Game B. If the game translates to the same cell values one-to-one, then Game A and Game B are really the same basic sudoku game, because they require the same logic to solve the puzzle.

Game A and Game B can be equated by color coding the Given Cells of the games to see if they match. If they match, then Game A and Game B are essentially the same sudoku game with different labels and possible rotations and reflections. 

It turns out that WebSudoku games appear to have a lot of repeats of Given Cells patterns, and then they also have those games match with a one-to-one correspondence between the numbers in each cell of both games. I am not making a claim that an exact match of Given Cells pattern will always leads to an exact match of two games having the same values, but that has been the overwhelming trend observed in this project with the games of WebSudoku.

There are other ways that games are transformed that are "quasi" the same games, 'cousins'? hah

It has been noticed that some games have common rows 4 and 5, and that rows 1,2 and 3 are also common except there may be a case where row 1 and row 2 are swapped and row 3 stays the same or all three rows, 1,2 and 3 get swapped, etc. Then when both games are played, both Game A and Game B are discovered to have the same basic sudoku game, except the rows swapping. The sequence of 'colors' is exactly the same from left to right in the corresponding rows of Game A and Game B. The swapping is symmetrical for the entire game, so that if row 1 and row 2 were swapped, then on the opposite side of the game, row 8 and row 9 were swapped also.

At the time of this rough draft preparation, I have recorded 581 game patterns of Given Cells, as well as several complete game plays and complete comparisons. Out of these 581, there have been found 24 game pairs that are the same through simple rotations and reflections and 84 games that are called 'Interesting Pairs or Triplets or more'. They are two games that can be equated by also performing row swapping of 2 or 3 rows of both 1,2,3 and 7,8,9 while leaving 4,5,6 intact. Several cases have been found where 3 or 4 games match each other through some combination of simple rotations, reflections and/or 'Interesting Pairs' treatment. 

132 total matches - 48 exact matches (24+24 Found)  = 84 interesting pairs 

So 132 * 100 / 581 = 23 percent matching games from a random sampling of 581 games 

Examples of Exact Match:

Example 001

Example 002

Example 003

Row Swapping

Example of 3 games that the same pattern of Given cells except for Row Swapping

The red and green lines between the games shows which rows are swapped; the others remain the same

Game Number Similarity

There was also discovered that the game numbers are 'fairly close' on the matches that have been found, so it does appear to be an intentional design effort by the game creators. Notice that the three games in the row swapping example are close to 5.2 billion, as well as the three games listed in the table for the exact game matching. I believe all of the 132 matches that I have found so far have their game numbers fairly close on a percentage of value basis.

The Future and a Peek at The Blossom Techniques

I will update this blog post when I have more information and plan to furnish a table of several examples. I also found another far more complicated form of game cousin that is more involved with exact swapping and rotating of parts of the game while other parts are static, that I have nicknamed 'Blossoms'. But the end result is still that Game A transforms into Game B in a systemic symmetrical manner in multiple cases. So far, two blossom pairs have been found in the first 581 games. 

The Blossom II Technique