Box elimination

Box elimination is a very common “candidate elimination” technique. You can remember it like this:

When a digit is “locked” at the intersection of a box and a line, the other side can’t contain that digit.

We use r1c1 to describe a cell position: r = row, c = column.

Tip: in the images below, green cells show the “locked positions”, and red candidates are the candidates you can remove in this step.

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Example 1: A column locks a digit into one box (remove the red candidates in that box)

First, look at the image below: focus on the highlighted column 4 and the two green cells r8c4 and r9c4.
You’ll notice that in column 4, digit 6 appears only in those two green cells.

This means: the 6 in column 4 must be placed in box 8 (because r8c4 and r9c4 are both in box 8).

Now look at the red 6 candidates inside box 8 — they can be removed.
A quick proof by contradiction:

• Assume one of those red positions in box 8 is actually 6
• Then r8c4 and r9c4 (in the same box) cannot be 6
• But then column 4 would have no place left for digit 6 — contradiction
  ⇒ so those red 6 candidates in box 8 are impossible and can be removed

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Example 2: A box locks a digit into one row (remove the red candidates in that row)

First, look at the image below: focus on the highlighted box 2 and the two green cells r1c4 and r1c5.
You’ll notice that in box 2, digit 9 appears only in those two green cells.

So the 9 in box 2 must be either r1c4 or r1c5.
And since both are on row 1, any red 9 candidate elsewhere on row 1 can be removed:

• Assume some other cell on row 1 is 9
• Then r1c4 and r1c5 (same row) cannot be 9
• But then box 2 would have no place for digit 9 — contradiction
  ⇒ so those red 9 candidates on row 1 are impossible and can be removed

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Summary

You don’t need to memorize extra names — just remember the two “directions”:

• From a row/column to a box: if all candidates of a digit in the line fall within one box, remove that digit from the rest of the box.
• From a box to a row/column: if all candidates of a digit in the box fall within one row/column, remove that digit from the rest of the line.