Claiming is similar to pointing, but works in reverse. When you know a candidate can only appear in one box within a particular column or row, you know that you can eliminate all other occurrences of that number within that box. With pointing the row/column is already restricted to one in that box and candidates are eliminated along the row or column in the other boxes, but with claiming you know that if a number can't appear in two of the boxes for a row or column, it must be in that row or column for the third (last) box and you can eliminate all other candidates from within the same box
Again the term "direct" here means the eliminations directly reveal a hidden single that you can immediately fill in, as we saw with direct pointing, and can generally be found without needing to write the candidates down. This is perhaps harder to spot than with pointing as with claiming you have to spot the candidates (or absence of them) along the whole width or height of the grid and back to a box, rather than spotting a "locked in" row/column within the smaller area of an individual box.
As is usually the case, an example will make this a lot easier to understand!
Look at the bottom 3 boxes (boxes 7, 8 & 9) and you will see that number 9 isn't a possible candidate for row 8 in boxes 8 and 9 - it can't be in the yellow squares in that row. That means '9' must be in row 8 in box 7 as it has to appear once in the row to satisfy the requirements of sudoku.
Now you know that the number 9 must be in that row within that box (the green squares) you can eliminate any other occurrence of that candidate that is within the same box but not in the same row (the red squares). You can now eliminate '9' from B7, C7 and C9 or in row/column notation we can write: 9 in r8 => r7c23,r9c3<>9. This says that because of the 9 in row 8, row 7 columns 2 & 3 and row 9 column 3 can't be the number 9. The 9 has "claimed" row 8 within box 7.
You'll now see that because C9 (r9c3) can no longer be a 9, this means G9 (r9c7) becomes a hidden single in row 9 - it's the only cell left in row 9 for the number 9 to go and we can directly fill it in!
The same principle applies to columns (the sudoku grid can obviously be rotated by 90° and still be a valid puzzle), and in this example we see that in column 3, the number 9 is not an option for boxes 1 and 7. This means that for column 3, '9' must be in box 4 and we can eliminate any occurrence of the candidate number within box 4 which isn't in column 3. This removes the '9' from B4 and A6, or in row/column notation: 9 in c3 => r4c2,r6c1<>9.
Now we have eliminated '9' from B4 we have directly revealed a hidden single in B1 / r1c2 as there is no other possible place for '9' in column 2.
The eagled-eyed among you may have spotted a "naked pair" in column 2 at B2 and B5, which eliminates '7' from B8 and reveals it as a number 5 naked single, but more about naked pairs in a later chapter!
Direct Pointing | Direct Hidden Pair
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