6's are determined the same way. the two 6's in the lower left and the lower middle, shows one position in the lower right elgible for a 6. The 6's in upper middle and lower middle show that a 6 goes in the middle middle square as shown. The 6's in the upper left and middle upper, show a 6 in the upper right square. The 6 in the upper middle eliminates all but one position in the middle middle square. And so on.
Now we can look for obvious solutions outside the box. The first thing to do is look for nearly complete solutions such as a column with only two numbers missing (8,9) for instance. Then by observation, you can tell that an 8 has to go in the lower right square (which already has a 9), leaving a 9 for the upper right square. All of the shade of grey numbers were determined in this fashion until not being able to progress any further.
What's left, is to try to find another singularity by observing the possible solutions for the remainder of the puzzle as shown by the small red numbers. The rows and columns and squares with the least open numbers are done first. This may go on for quite a few numbers or if you are lucky, a few numbers until a singularity is revealed. Notice for instance in the upper left square, the numbers 8,9 are unique to those two positions and cannot contain any other numbers in those two positions
leaving the missing numbers 2,7 for the remaining two positions. It is important to remember that when two numbers exist like this, they can be used as part of the solution as well.
The green numbers are a result of finally finding a singularity by the above method and then leveraging this to determine the next number. The above puzzle is nearly complete and easily finished.
To check if your solution is valid, look for and determine that each row and column do not contain any duplicate numbers. You can also wait for "tomorrow" to get the solution to "yesterdays" puzzle (boring).