Up to now I've focused on solution rules in isolation, always assuming that a complete set of "pencil marks" (candidate numbers for a cell) are available. For a computer, this is no problem. But when solving a puzzle with pencil and paper, it's useful to have some strategies that can allow you to make progress without requiring all those marks.
Today I'd like to talk about what I like to call "eyeballing" (that is, solving just by looking at things) - filling in numbers based on elimination along rows and columns. Every 3x3 box has 4 other boxes that influence it. For example, the top middle box is influenced by the two top corner boxes to its left and right, and the two boxes below it:
A value in any of those 4 "influencing" boxes will let use eliminate that value from the portion of the box that occupies the same row or column. In the best case, we might have four clues that, taken together, only leave one possible cell that can hold the value. In the diagram below, 4 X's together tell us the only possible location of the X in box 2.
What's really going on here? This is just the rule of uniqueness looked at in a slightly different way. We can fill in the X in box 2 because there is only one possible cell that can hold it.
Of course, there are rarely going to be 4 values so strategically placed. Fortunately, if some cells are already occupied by clues and previously filled-in values, we can learn something from only 1 or 2 values in "influencing" boxes:
Since two boxes are already occupied by clues ("A" and "B"), the two X's in the corner boxes give us enough information to place the middle X. In general, we want to look for common values in the 4 related boxes,
so that there is a chance that we can cross out all but one cell of the
box we're interested in. We want to pay extra attention to "crowded" areas of the box where just a few values might be enough to let us cross out all but one cell.
Let's solve Thursday's puzzle step by step, starting with eyeballing some values. Here are the clues:
We see a lot of 2's in the boxes that influence box 1. Let's see what happens when we cross out all of the rows and columns that contain 2's. Only one space is left in box 1, so the 2 must go there:
We can do the same for 7's surrounding box 2:
Now we use the 8's in the boxes below box 3 to locate the 8 in that box:
This type of "eyeballed" value is particularly easy to find - both values lie in the same column of boxes so we don't have to look in two directions.
We can use the new values we've just filled in to help us eyeball additional values. Since we filled in a 7 in box 2 a few moments ago, now we have enough information to eyeball the 7 in box 3.
This 7, in turn, lets us eyeball the 7 in box 9:
See if you can eyeball any more values. Two possibilities are shown below.
Note that the two 9's in the left columns would have given us enough information by themselves to place the 9 in box 1. Because column 3 of box 1 already has two values filled in, we don't need to look in the boxes to the right for help.
There are at least three more values in this grid that can be eyeballed. Probably there are a few more that I haven't managed to spot.
Next time I'll show how eyeballing can give us information even when we can't fill in a number right away.