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Counting sudokus (Posted on 2018-07-13) Difficulty: 3 of 5
You are given a 9x9 grid of numbers such that:
a. each row,
b. column,
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 ?

No Solution Yet Submitted by Ady TZIDON    
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Solution re: solution (part 1 only) | Comment 3 of 6 |
(In reply to solution (part 1 only) by Steven Lord)

I can see the logic and don't see any flaws.

However, Sloane's OEIS 

A002860 Number of Latin squares of order n; or labeled quasigroups. 

shows the 9th element as


That's about 5.5 x 10^27

I don't see where the extra possibilities come from.


Now I do see where the extras come from.

Say you are placing the 3's. According to the logic presented here, you would have a choice of 7 places in the top row (unoccupied by either 1 or 2), 6 places in the second row (unoccupied by 1 or 2 and not under a preceding 3), etc.  However in some instances a preceding row's 3 may be in the same column as one already restricted by 1 or 2, so that does not further reduce the available columns. This gets more prevalent as you get further down and as you get to higher numbers.

Edited on July 14, 2018, 3:51 pm
  Posted by Charlie on 2018-07-14 15:35:08

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