Last week a box pointed to a line. This week a line claims a box. It is the same fact read from the other end, and it is the second of the two Yellow Belt lessons. Learn it and you have both halves of what an intersection can prove.
A line claims a box
Take one row and one digit. Mark every square in the row where that digit can still go. If all of those squares sit inside the same box, the row has to place the digit in that box. A box holds each digit once. So the box has spent its copy of that digit on the row, and every square in the box that lies off the row loses it. Columns work the same way.
Here is a real position. Wednesday's Medium, No. 72, exactly as it printed, with 40 given digits and nothing written in yet.
Row 2 holds three of them: a 2 at column 2, a 7 at column 3, a 3 at column 6. Six squares are empty, at columns 1, 4, 5, 7, 8 and 9.
Where can a 9 go in row 2? Not at column 1, which already holds a 9 at row 6. Not at column 7, where row 1 holds one. Not at column 8, where row 5 holds one. Not at column 9, where row 7 holds one. Four of the six squares are gone. Row 2 column 4 and row 2 column 5 are what remain.
Both of those sit in the top middle box, rows 1 to 3 and columns 4 to 6. That box has five empty squares: row 1 column 5, row 2 columns 4 and 5, and row 3 columns 4 and 5.
The 9 of row 2 is at one of the two squares in row 2, so it is inside that box either way. The box cannot put a 9 anywhere else. Row 3 column 4 was showing 4, 5 and 9. Row 3 column 5 was showing 1, 4, 5 and 9. Both lose the 9. Row 1 column 5 never had one, since row 1 holds its 9 at column 7.
Now look down column 4. Row 1 column 4 is a 7, row 4 column 4 is a 2, row 9 column 4 is a 1. Of the squares still open, row 5 holds its 9 at column 8, row 6 holds its 9 at column 1, row 7 holds its 9 at column 9, and row 8 holds its 9 at column 6. Two squares in the whole column could take a 9: row 2 column 4 and row 3 column 4.
Row 3 column 4 has just lost its 9. So row 2 column 4 is a 9, and it is certain.
That is the move paying for itself. One elimination in row 3, and a digit is proven in row 2.
The usual mistake is to keep clearing along the row. Nothing in the row has been proven. Row 2 columns 4 and 5 keep every candidate they were holding, and the rest of row 2 is untouched. Every elimination happens inside the box, on the squares that lie off the line.
Solving references give this move more than one name. Hodoku files it as claiming, or locked candidates type 2, with last week's pointing pair as type 1. It is one idea in two directions, which is why the two lessons sit together.
The week's editions
Seven printed: Easy No. 70 and No. 71 on Monday and Tuesday, Medium No. 72 and No. 73 on Wednesday and Thursday, Hard No. 74 on Friday, Master No. 75 on Saturday, Expert No. 76 on Sunday.
The first five have something in common that the labels do not show. Between them they hold 195 empty squares, and every one of those squares comes out with a full house or a naked single. Not one hidden single is needed across the five days, and no pencil marks either. Friday's Hard prints with 31 givens and 50 squares to fill, and it is still all singles.
Saturday's Master, No. 75, is where that stops. It prints with 25 givens, and the singles run out partway down. The move that starts it again is a pointing pair. Sunday's Expert asks for more than that.
In Claire's Sudoku
Readers finished 51 puzzles over five days and used a hint on three of them. Over the same five days a week earlier, 50 puzzles took 19 hints. Wrong entries came down from 92 to 19 across almost the same number of boards, at much the same mix of levels. Nine hard boards went out and all nine were finished. The week before, five went out and two came back finished.
Box-line reduction is the second lesson of the Yellow Belt and the one that finishes the belt. If you do one thing this week, open that lesson in Claire's Sudoku and work it through once.
The world championship in Kolkata starts next month. I said I would write about it once the results are in, and I will.