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Strategies for Popular Number Puzzles

Law of Leftovers

There is an excellent strategy for Jigsaw Sudoku it is vital to know about - and it uses the odd shapes of the boxes.

Figure 1: Leftover Region
Figure 1: Leftover Region
Consider the area marked with As in Figure 1. It consists of three cells in the stair-shaped “box”. All the solution numbers for these three cells cannot appear in the cells marked B, since A and B share the top two rows. This means that all As must be in the cells marked C.

The top two rows are referred to as a “region” while the cells marked C are “leftovers”.

A region can be a single row or column, two adjacent rows or columns (as on the right), or even three or four rows or columns – although there is no significant gains from four or more in a region. A double row or column seems to be the most common, and in such cases the Law of Leftover is as useful as Intersection Removal.
For the Law of Leftovers to work, we want to find a region with an overlapping box – the leftover being the part of the box hanging outside the region. However many cells are left over, we need to find another box that also overlaps our region by the same number of cells. CCC and AAA in Figure 42.1 satisfy this requirement. The examples below should convince you that it’s worthwhile to look out for such patterns.

Figure 2: Leftover Region Examples
Figure 2: Leftover Region Examples

If we examine one possible Sudoku solution for our grid, shown in Figure 2, we can see, as if by magic, that 1/2/6 occupies the areas marked AAA and CCC (areas marked in yellow). That example seems to work. It also works if you take the last two columns on the right.

There is another region at the bottom of the jigsaw – the last three rows marked with a blue ring. Cells F2/F3/F4 contain 9/6/2. The central “I”-shaped box has three cells inside the region; this must be our counter-partner and indeed it contains 6/2/9. This knowledge is going to be very useful.
There are a number of ways to think about this trick. We know that all the numbers in a row, a column or a box always add up to 45, since the numbers 1 to 9 occur once in each unit. If we have a box that is partially within the region, it will contain a certain set of numbers outside the region. In order for the 45 rule to be satisfied for all boxes as well as rows and columns, any deficit must be made up elsewhere – with identical sets of numbers. With two rows or columns, the sum we need to have in mind is 45+45=90. Likewise for three rows or columns, the sum is 135.

The usefulness of this strategy depends on how helpful the design of the jigsaw is. Some jigsaws will be well-suited to it, while others will be very mean, in that you might not find a single instance.

Figure 3
Figure 3 : Load Example or : From the Start
You’ll want to check for LOL occasions just so you don't need to add notes for certain cells. Let’s look at this example of a two-row region at the bottom of the grid. The "outies" are marked in green and correspond to the "innies" in brown. The Law of Leftovers implies that the two numbers in the green cells must be the same two numbers in the brown cells - although it is not possible to know which way round yet,

However, we have 3, 6, 8 and 9 as possibles so far. 3 has been eliminated from the brown squares and 9 has been eliminated from the green cells - so these are not candidates for both sets. We are left with 6 and 8 as the only possible solutions for the leftover regions. That's major progress since we get a 6 in G3 and an 8 in H1.
Figure 4
Figure 4 : Load Example or : From the Start

As we saw in the previous example, the leftover region - the bit that sticks out, can be made of cells from more than one shape. The leftover region must be all the cells that stick out (green cells). There will always be an equal number of cells from outside shapes poking in (brown cells).

In this example, three cells marked in green extend out of the top three rows and form the leftover. Three brown cells intrude into the top three rows. We can ask - "what are the common candidates". This tells us that the 8 in D5 can be eliminated.
Figure 5
Figure 5 : Load Example or : From the Start

The final example shows a 5-cell left over region. We need five numbers to fill these five cells and column 3 provides them: they must be 3, 5, 6, 7 and 9. Our "innie" area of brown cells has all these numbers, but also a profusion of 4s. They can be removed.

That gives us a a Hidden Pair on 4 in F4 and G4, so we can progress by removing the 4 in E4.



... by: ralph maier

Friday 16-Jun-2023
in todays puzzle 5239 since i donot have a printer ialways try to solveiit by using the law of leftovers. It is solvable so iwonder why the solver uses the law of leftovers so seldom?? ialso use triple pointing cells etc
Andrew Stuart writes:
L of L is second to Double Pointing Pairs which seem to attack the same candidates
ralph maier replies:Friday 16-Jun-2023
Dear Andrew. I am well aware where LL is placed. First of all you doing a great job in advancing knowledge about puzzles. I was simply commenting that the solver uses the LL seldom. After I solve a puzzle sudoku jigsaw killer straights I always check how the the solver does it. So about jigsaw I have very seldom seen it to use it. I have solved quite a few jigsaw puzzles using only logic and LL without using AIC cycles Medusa etc. So I was wondering why the solver does not use LL more often. Regards and continue the great JOB YOU DOING ON YOUR SITE
Andrew Stuart writes:
It is the order the solver looks. Try unchecking Double Pointing Pairs in the solver and LL should show up
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... by: Hanbo Kang

Saturday 7-Mar-2020
I have noticed the figure 3, where 3s in the G3 and the G7 cell can be probably eliminated by the rule of the double line/box reduction. (see the H, J rows). Likewise we could erase the two 9s by means of the double pointing pairs.

... by: Eldora

Wednesday 26-Dec-2012
Wonderful explanation of facts available here.

... by: David N. Jansen

Monday 30-Apr-2012
I think in the final example also a 1 would be allowed (see cells G3 and DF2).

... by: ahmet çilek

Saturday 5-Dec-2009
thanks for this site. can you add "design new jigsaw sudoku" ?
Andrew Stuart writes:
In my job queue. Not as simple as it sounds though!
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Article created on 13-April-2008. Views: 77282
This page was last modified on 13-February-2012.
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