When it comes to Jigsaw Sudoku, it can't be emphasised too much how important Pointing Pairs and Line/Box Reduction are for candidate removal and general puzzle cracking. They are important for normal Sudoku puzzles but excel in Jigsaws because the shapes allow for many more instances. And not just Pairs and Triples but Quads and upwards.

This extension is very easy to spot - so I have put it before Law of Leftovers in the solver. Experienced Jigsaw solvers are probably using this all the time!

The strategy family of**Intersection Sets** looks at how rows and columns intersect or overlap with boxes (or shapes in Jigsaw). It is fascinating to know that precisely because Jigsaws are very odd shapes - often elongated, we can improve the *one box/one line* Intersection Removal technique just for Jigsaws. We can 'double up' using two boxes and two lines (two rows or two columns).

Pointing Pairs and Line/Box Reduction are flip opposites of each other, and same goes for their double versions. Pointing Pairs look for a box containing candidates X that all line up on a row or column. The solution must exist in the box so it excludes X from the rest of the line or column. What if we find two boxes where all the candidates line up on exactly two rows (or two columns)? Then we know any other candidate*pointed at* can't be the solution.

This extension is very easy to spot - so I have put it before Law of Leftovers in the solver. Experienced Jigsaw solvers are probably using this all the time!

The strategy family of

Pointing Pairs and Line/Box Reduction are flip opposites of each other, and same goes for their double versions. Pointing Pairs look for a box containing candidates X that all line up on a row or column. The solution must exist in the box so it excludes X from the rest of the line or column. What if we find two boxes where all the candidates line up on exactly two rows (or two columns)? Then we know any other candidate

Here is a clear example. All the green highlighted 2s exist in just those two columns AND in just those two shapes. We don't know where in those eight possible places the two solutions for 2 must go - but it excludes 2 from the rest of the columns outside the two shapes. The solve returns:

Double Pointing Pairs

The two boxes 1 and 4 contain 2 in just columns 1 and 2, so...

- 2 can be removed from H1

- 2 can be removed from J1

- 2 can be removed from G2

- 2 can be removed from H2

- 2 can be removed from J2

It's worth pausing at this point to consider why this doubled up version doesn't work in normal Sudoku. In that puzzle you never get more than three boxes overlapping any row or column - the nine 'shapes' are arranged specifically in that symmetrical way. Take a look at cells H4 and J4. They belong to the shape in the bottom left corner but stick way out into the fourth column. At least two numbers from 1 to 9 must go in there, so whatever they are, they will not be in the rest of the shape. I can tell from the result of this strategy that 2 has been excluded from all the yellow highlighted cells and 4 is a clue in J3. So I am pretty certain 2 will be the solution of one of those two cells. The 2 in B3 satisfies the requirement for column - but is in a completely different shape. These innies and outies that poke into columns and rows don't exist in normal Sudoku, so you'll never get the opportunity for this arrangement except in a Jigsaw puzzle.

Immediately following the elimination above we get another Double Pointing Pairs. Not every sequence will be this generous; sometimes only one or two candidates can be removed, but it might save you a more complex strategy down the line.

Double Pointing Pairs

The two boxes 2 and 3 contain 4 in just rows A and C, so...

- 4 can be removed from A1

- 4 can be removed from A2

- 4 can be removed from C1

- 4 can be removed from C2

- 4 can be removed from C7

I've chosen this puzzle as it contains a good number of normal Intersection Removal and many of the doubled up version. I suggest you run it through the solver from the start.

In this example I wanted to show how the two lines do not have to be adjacent. They can be as far apart as the greatest extent of the two boxes (as long as they share rows or columns). In this case there are two rows between. The solver gives this result:

Double Pointing Pairs

The two boxes 2 and 4 contain 2 in just rows B and E, so...

- 2 can be removed from E5

- 2 can be removed from E6

Again, this is a good example of a "tough" grade puzzle with many Intersection Removals.

By doubling up the first strategy I'm begging the question, can you triple up as well? There is no logical reason why not, but this is rare as it would require five different shapes to intersect in a row or column. I have chosen not to implement it in the solver as I'd need to insert much further down the list of strategies but at some future point I will code it up and see if it is useful at all.

If you work out an example, please let me know and I'll add it to this page.

(The word "Pair" in this strategy name strains credulity since it will be a rare occasion when exactly two pairs are involved. The limit is this length or width of a Jigsaw shape and often there will be different numbers involved in each "pair". "Set" is perhaps a better word, but I retain "pair" because I want to associate it with the single one line/one box version where pairs are most common).

Double Pointing Pairs are only half the story. Do read the next article on Double Line/Box Reduction for the complete picture.

I am grateful for

## Comments

Comments Talk## Friday 16-Jul-2021

## ... by: Steven

The July 16 Daily Jigsaw has an example of a triple pointing pair## Monday 15-Feb-2016

## ... by: Brian Hobson

I do believe that Multiple (i.e., Triple, Quadruple and Quintuple) Pointing Pairs and Multiple Line/Box Reduction have made the Law-of-Leftovers redundant. i haven't come across a case of L-o-L for years. They have all been solved before i get to them by the above strategies.## Monday 14-Apr-2014

## ... by: puzzler77

In the first example, the law of leftovers indicate the 2 found in B3 must appear in either H4 or J4 eliminating 2 from each of the five cells mentioned, however, for some, the double pointing pairs will be easier to see.