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Intersection Removal
From sudokuwiki.com, the puzzle solver's site
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If any one number occurs twice or three times in just one unit (any row, column or box) then we can remove that number from the intersection of another unit. There are four types of intersection:
  1. A Pair or Triple in a box - if they are aligned on a row, n can be removed from the rest of the row.
  2. A Pair or Triple in a box - if they are aligned on a column, n can be removed from the rest of the column.
  3. A Pair or Triple on a row - if they are all in the same box, n can be removed from the rest of the box.
  4. A Pair or Triple on a column - if they are all in the same box, n can be removed from the rest of the box.
Rules 1 and 2 are also called Pointing Pairs/Triples Rules 3 and 4 are also called Box/Line Reduction


Type 1 - Pointing Pairs/Triples Strategy (a.k.a. Intersection Removal)

Looking at each box in turn there may be two or three occurrences of a particular number.
If these numbers are aligned on a single row or column (as a pair or a triple) then we know that number MUST occur on that line. Therefore, if the number occurs anywhere else on the row or column outside the box WHICH THEY ARE ALIGNED ON then it can be removed. The pair or triple points along the line at any numbers which can be removed.

Pointing Pairs Example 1

Consider the top third of the puzzle board above. We are looking at the number 2 in the center box and using the solver all the cells with 2 remaining have been highlighted. 2 can only be found in the centre row at B4 and B6. The 2's in the rest of the row can be removed (red squares).

Now this is a rather special puzzle and a little extreme, but if we look at the whole board you can see I have highlighted a whole cluster of Pointing Pairs. It is obviously not necessary to spot everyone to progress the board but there are so many good examples it is worth looking at. The eliminations are highlighted in yellow. You should be able to see which eliminations belong to which Pointing Pair.
Pointing Pairs Example 2
Pointing Pairs Example 2: Load Example
Type 2 - Box Line Reduction Strategy (a.k.a. Intersection Removal)

This strategy involves careful comparison of rows and columns against the content of boxes (3 x 3 squares). If we find numbers in any row or column that are grouped together in just one box, we can exclude those numbers from the rest of the box. For example:

Box Line Reduction Example

Consider the right hand box in this center row of the board. We can see five squares with 9s marked as possible numbers. The 9s that exist in cells A and B are the only 9s in that whole row. Either A or B MUST contains a 9 in the final solution. We can therefore safely remove the 9s from C, D and E.


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Comments...

Wednesday 7-Apr-2010

... by: alvin

NICE SITE.. ive been to many sites ( videos, explanation, sodoku solver..) this site is the best of the best... thanks man. ^_^

Thursday 25-Feb-2010

... by: CS VIDYASAGAR

Another excelleng explaination of difficult concept. In Type I , digit 2 have no alternative as it has to be in cell B4 and B6 in the Centre Box. Where as in Right Box, digit 2 appear in Row B and Row C. Hence the Centre Box has priority to contain digit 2. Since 2 has to be either in cell B4 or B6, it has no other place to go in the same Row ie. Row B. So 2 from box Right in Row C can be removed. The concept is NO ALTERNATIVE FOR DIGIT IN ROW OR COLUMN IN A PARTICULAR BOX.
Thanks for nice and simple elucidation of difficult concept.

Wednesday 20-Jan-2010

... by: Patrick Barnaby

These pairs are easy to spot if you first look at a box then ask is there a single line in a box? But to spot the sevens and eitghts you have to see the two X-Wings first.

There is an X-Wing for sevens and an X-Wing for eights.

Friday 20-Nov-2009

... by: Clell

It would have been much easier to understand if you had made the point of alternatives. I did not understand why the double 2s could not eliminate the triple 2s and vice versa until I figured out that what matters is whether or not there is an alternative choice. GAR.

I had looked at this and also could not see the relationship until I reread the above and saw there were no other alternatives in the pointing pair box that forced the others to be eliminated. thanks


Friday 25-Sep-2009

... by: Greg

Moses - If A or B is a 9, then how can there be another 9 in the rest of those box of nine squares? You can therefore eliminate the possibility of a 9 in C, D or E.

Thursday 17-Sep-2009

... by: grosenthal08@cox.net

It would have been much easier to understand if you had made the point of alternatives. I did not understand why the double 2s could not eliminate the triple 2s and vice versa until I figured out that what matters is whether or not there is an alternative choice. GAR.

Sunday 30-Aug-2009

... by: W Gerald

Why isn't your printed version the same as the site version with no changes in layout and only background and text colours reversed? The site version display is superb.

Andrew Stuart writes:
It should be identical. It removes the side menu - not required for printing - and it horizontally flexes since the printer driver decides on the actual width.

Sunday 12-Jul-2009

... by: Rick Aben

Shouldn't this strategy be called "intersection prevalence" (or something like that) instead of "intersection removal", because what you actually do (see for instance example nr 1) is keeping the two 2's in the intersection of row B and box 2 (since they do not appear in the remainder of box 2, so that one of them must be a solution in that box) and remove the 2's out of the remainder of row B (in box 3).
The same counts for "type 2" (with a swap of box and rows). In that example the two 9's can be found in the intersection of row 4 en box 6 (the box on the right) and not in the remainder of row 4, which means the 9's in the remainder of box 6 can be eliminated. Ofcource, the solution is the same, the difference is in the logical explanation. Isn't it?

Andrew Stuart writes:
The name stuck - in the early days. Not my invention but you are perhaps closer to the mark

Sunday 21-Jun-2009

... by: moses

please explain more on box line reduction and show some examples thanks

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Article created on 3-April-2008. Views: 89986
This page was last modified on 12-April-2008, at 12:12.
All text is copyright and for personal use only but may be reproduced with the permission of the author.
Copyright Andrew Stuart @ Scanraid Ltd, 2008