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The '''Law of Leftovers''' is a [[solving technique]] that operates on [[Sudoku Variations]] with irregular [[region]]s, such as [[Jigsaw]] Sudoku. The technique was devised by Bob Harris.

The technique is based on the fact that an equal number of [[house]]s must have an equal number of occurrences for each [[digit]].
== Selecting Law of Leftovers regions ==
It may be difficult at first to find the best spots to apply this technique, but there is a method to quickly identify the most promising areas in the grid. Just look for borders between rows or columns which largely coincide with jigsaw borders. In the example, the border between column 6 and 7 has 5 cell borders which are also the border of jigsaw regions.

[[Image:LOL1.png|Law of Leftovers example 1|right]]
* Draw a line separating a certain number of jigsaw regions.
: In the example, the red line separates the 3 light green jigsaw regions.
* Draw a line separating the '''same''' number of rows or columns.
: In the example, the blue line separates the left 3 columns.
* Identify the parts of the jigsaw regions '''outside''' the separated rows or columns.
: The example below has 2 cells sticking out.
* Identify the parts of the remaining regions '''inside''' the separated rows or columns.
: In the example, these parts have a light blue color.
* Verify that the number of ''innies'' and ''outies'' are the same.
: There are 2 green outies and 2 blue innies in the example.
* Compare the innies and the outies.
: See the next section for Law of Leftovers logic.

== Comparing the innies and the outies ==
In the example, we see that one of the outies ('''r4c6''') contains digit 8. The green cells inside the last 3 columns only have room for 2 more instances of digit 8. The innies must therefore supply the missing digit 8. Fortunately, '''r3c7''' already contains digit 8, so this condition is satisfied.

The other outie contains digit 5. Again, the blue innies must compensate this loss. There is only one innie left, so we can place digit 5 in '''r9c7'''.

The '''Law of Leftovers''' states that '''in the solution, the innies and the outies contain the same digits'''. From this law, we can make the following deductions:
* A digit placed in an outie must also be placed in one of the innies.
* A digit placed in an innie must also be placed in one of the outies.
* A digit for which no outie has any candidates can be eliminated from all innies.
* A digit for which no innie has any candidates can be eliminated from all outies.
* When all innies belong to the same [[line]], a digit which must be placed in one of the innies can be eliminated from the other cells on that line.
* When all outies belong to the same line, a digit which must be placed in one of the outies can be eliminated from the other cells on that line.

== External links ==
* [http://www.sudocue.net/forum/viewtopic.php?t=656 Examples of Law of Leftovers]
* [http://www.bumblebeagle.org/dusumoh/8x8/tutorial/index.html Original Law of Leftovers tutorial]
* [http://www.djape.net/sudoku/wp/index.php/2007/08/24/the-law-of-leftovers-lol/ Law of Leftovers explanation at Djape.Net]
[[Category:Solving Techniques]]

Law of Leftovers - Revision history

Rooted: Created page with "The '''Law of Leftovers''' is a solving technique that operates on Sudoku Variations with irregular regions, such as Jigsaw Sudoku. The technique was devised..."