See the article above for more details.
Broadly speaking, the method just looks at all possible grids and counts which ones are Sudoku grids. Of course, since this would take prohibitively long, even with the excellent programs of Bertram and Ed, it is necessary to simplify the search a little. Let's try to indicate how we do this.
We break up the search into two steps.
In fact, the actual number of possible configurations for the top three rows is 948109639680! However, we don't have to do the second step for all of these! For example, if we change all the digits 1 to 2 and vice versa, then any way to complete a grid for one of the configurations can be adapted to a way to complete a grid for the other; just replace all 1 with 2 and vice versa throughout the whole grid. In fact, any two configurations which can be got by relabelling the numbers are going to have the same number of completions in the second step. There are lots of possible relabellings like this (in fact, there are 362880 relabellings of any grid). And if you think further, you may be able to come up with other ways to transform the top three rows of a grid in such a way that the answer to the second step is unchanged (and so only needs to be counted once).
In fact, as the article shows, there are perhaps more of these transformations than you might expect. It turns out that every configuration of the top three rows gives the same answer in the second step as one of just 44. That is, each of the 948109639680 possible top three rows can be transformed by a sequence of these transformations into one of the list of 44. So we only need to work out the number of ways that each of these 44 possibilities can be completed to a full grid. At this point, clever computing can count this number in just a couple of minutes for each of the 44 possibilities.
Here are the results, taken from Ed Russell's program. Some words of explanation are in order. After relabelling, the first 3x3 block will always be taken to be
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
Now we list the 44 essential cases that we need to count. In order to complete the bottom six rows of a grid, we just need to know which numbers are in which columns of the top three rows, and their order doesn't matter. Since we already know the numbers in columns 1-3, we just list below the numbers in columns 4-9, in numerical order. The eighth column gives the number of the original configurations which have the same number of completions, and this number is given in the final column. (Note that the numbers in the eighth column need to be multiplied by 1881169920 to get the actual number; this accounts for relabelling and some other transformations which always exist.)
| Number | Column 4 | Column 5 | Column 6 | Column 7 | Column 8 | Column 9 | Number of equivalent configurations | Number of completions to a full grid |
|---|---|---|---|---|---|---|---|---|
| 1 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,5 | 3,6,7 | 4,8,9 | 2484 | 97961464 |
| 2 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,5 | 3,6,8 | 4,7,9 | 2592 | 97539392 |
| 3 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,5 | 3,6,9 | 4,7,8 | 1296 | 98369440 |
| 4 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,5 | 3,7,8 | 4,6,9 | 1512 | 97910032 |
| 5 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,6 | 3,4,8 | 5,7,9 | 2808 | 96482296 |
| 6 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,6 | 3,4,9 | 5,7,8 | 684 | 97549160 |
| 7 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,6 | 3,5,7 | 4,8,9 | 1512 | 97287008 |
| 8 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,6 | 3,5,8 | 4,7,9 | 1944 | 97416016 |
| 9 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,6 | 3,5,9 | 4,7,8 | 2052 | 97477096 |
| 10 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,7 | 3,4,8 | 5,6,9 | 288 | 96807424 |
| 11 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,7 | 3,5,8 | 4,6,9 | 864 | 98119872 |
| 12 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,8 | 3,4,7 | 5,6,9 | 1188 | 98371664 |
| 13 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,8 | 3,5,7 | 4,6,9 | 648 | 98128064 |
| 14 | 1,2,4 | 3,5,7 | 6,8,9 | 1,2,8 | 3,6,9 | 4,5,7 | 2592 | 98733568 |
| 15 | 1,2,4 | 3,5,7 | 6,8,9 | 1,3,5 | 2,6,9 | 4,7,8 | 648 | 97455648 |
| 16 | 1,2,4 | 3,5,7 | 6,8,9 | 1,3,5 | 2,7,8 | 4,6,9 | 360 | 97372400 |
| 17 | 1,2,4 | 3,5,7 | 6,8,9 | 1,3,6 | 2,5,9 | 4,7,8 | 3240 | 97116296 |
| 18 | 1,2,4 | 3,5,7 | 6,8,9 | 1,3,8 | 2,6,7 | 4,5,9 | 540 | 95596592 |
| 19 | 1,2,4 | 3,5,7 | 6,8,9 | 1,3,8 | 2,6,9 | 4,5,7 | 756 | 97346960 |
| 20 | 1,2,4 | 3,5,7 | 6,8,9 | 1,4,5 | 2,6,9 | 3,7,8 | 324 | 97714592 |
| 21 | 1,2,4 | 3,5,7 | 6,8,9 | 1,4,5 | 2,7,8 | 3,6,9 | 432 | 97992064 |
| 22 | 1,2,4 | 3,5,7 | 6,8,9 | 1,4,6 | 2,3,9 | 5,7,8 | 756 | 98153104 |
| 23 | 1,2,4 | 3,5,7 | 6,8,9 | 1,4,7 | 2,6,9 | 3,5,8 | 864 | 98733184 |
| 24 | 1,2,4 | 3,5,7 | 6,8,9 | 1,4,8 | 2,6,9 | 3,5,7 | 108 | 98048704 |
| 25 | 1,2,4 | 3,5,7 | 6,8,9 | 1,5,6 | 2,3,9 | 4,7,8 | 756 | 96702240 |
| 26 | 1,2,4 | 3,5,8 | 6,7,9 | 1,2,5 | 3,6,8 | 4,7,9 | 516 | 98950072 |
| 27 | 1,2,4 | 3,5,8 | 6,7,9 | 1,2,6 | 3,4,8 | 5,7,9 | 576 | 97685328 |
| 28 | 1,2,4 | 3,5,8 | 6,7,9 | 1,2,7 | 3,5,8 | 4,6,9 | 432 | 98784768 |
| 29 | 1,2,4 | 3,5,8 | 6,7,9 | 1,3,7 | 2,6,9 | 4,5,8 | 324 | 98493856 |
| 30 | 1,2,4 | 3,5,8 | 6,7,9 | 1,4,7 | 2,5,8 | 3,6,9 | 72 | 100231616 |
| 31 | 1,2,4 | 3,5,8 | 6,7,9 | 1,4,7 | 2,6,9 | 3,7,8 | 216 | 99525184 |
| 32 | 1,2,4 | 3,5,8 | 6,7,9 | 1,5,6 | 2,3,7 | 4,8,9 | 252 | 96100688 |
| 33 | 1,2,4 | 3,5,9 | 6,7,8 | 1,2,7 | 3,5,6 | 4,8,9 | 288 | 96631520 |
| 34 | 1,2,4 | 3,5,9 | 6,7,8 | 1,2,7 | 3,5,9 | 4,6,8 | 864 | 97756224 |
| 35 | 1,2,4 | 3,5,9 | 6,7,8 | 1,4,7 | 2,5,8 | 3,6,9 | 216 | 99083712 |
| 36 | 1,2,4 | 3,5,9 | 6,7,8 | 1,4,7 | 2,6,8 | 3,5,9 | 432 | 98875264 |
| 37 | 1,2,4 | 3,6,9 | 5,7,8 | 1,2,5 | 3,6,9 | 4,7,8 | 216 | 102047904 |
| 38 | 1,2,4 | 3,6,9 | 5,7,8 | 1,2,7 | 3,6,9 | 4,5,8 | 144 | 101131392 |
| 39 | 1,2,4 | 3,6,9 | 5,7,8 | 1,3,5 | 2,6,7 | 4,8,9 | 324 | 96380896 |
| 40 | 1,2,4 | 3,6,9 | 5,7,8 | 1,4,7 | 2,5,8 | 3,6,9 | 108 | 102543168 |
| 41 | 1,2,4 | 3,7,9 | 5,6,8 | 1,4,6 | 2,3,9 | 5,7,8 | 12 | 99258880 |
| 42 | 1,2,6 | 3,4,8 | 5,7,9 | 1,3,5 | 2,4,9 | 6,7,8 | 20 | 94888576 |
| 43 | 1,2,6 | 3,7,8 | 4,5,9 | 1,4,7 | 2,5,8 | 3,6,9 | 24 | 97282720 |
| 44 | 1,4,7 | 2,5,8 | 3,6,9 | 1,4,7 | 2,5,8 | 3,6,9 | 4 | 108374976 |
To get the final total, multiply the green number by the red number in each row, and take their sum. Then multiply by 1881169920 to get the final answer, which is 6670903752021072936960 in total.