You are given a 9x9 grid of numbers such that:
a. each row,
c. each of the nine 3x3 grids
has each number 1 through 9 exactly once.
Lets call it Sudoku puzzle's solution or SPS.
A grid with condition c. removed is called a Latin Square, LS.
1. (d2) how many 9x9 LS's are there?
2. (d4) And how many 9x9 SPS's ?
The first known solution to complete enumeration was posted by QSCGZ [Guenter Stertenbrink] to the rec.puzzles newsgroup in 2003, obtaining 6,670,903,752,021,072,936,960 (6.67×1021) distinct solutions.
In a 2005 study, Felgenhauer and Jarvis analyzed the permutations of the top band used in valid solutions. Once the Band1 symmetries and equivalence classes for the partial grid solutions were identified, the completions of the lower two bands were constructed and counted for each equivalence class. Summing completions over the equivalence classes, weighted by class size, gives the total number of solutions as 6,670,903,752,021,072,936,960, confirming the value obtained by QSCGZ. The value was subsequently confirmed numerous times independently. A second enumeration technique based on band generation was later developed that is significantly less computationally intensive.
Posted by Charlie
on 2018-07-14 15:42:49