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Strategies for Popular Number Puzzles

Pattern Overlay Method

Invented by Myth Jellies, this strategy looks at the way candidates of a specific digit N can be distributed in the remaining spaces. Every time a digit is placed it removes other spaces in the rest of the row, column and box, quickly narrowing down the possibilities. It is a strategy you don't want to apply too early in the puzzle since the number of overlays might be too large, but in the middle and end games it is fairly easy to apply.

The First Pattern Overlay
The First Pattern Overlay

The first diagram shows a possible pattern or template. It is in fact the first such pattern given an empty board and placement from top left to bottom right. On an empty board there are 46,656 different patterns which is why we use this when most cells are filled. Every placement of N reduces the number of patterns by a factor of 9.
Just the 3s shown
Just the 3s shown
In this relatively simple example all the 3s are shown. We can start from the top block which contains just two threes, so the total number of overlays will be two.
The two possible patterns
The two possible patterns
I have coloured the two patterns here. Try and find another pattern which picks a 3 for every row, columns and box. It should be impossible.
The Overlay
The Overlay
It helps to label the patterns "a", "b", "c" and so on against the candidate number. In this case we are only looking at number 3 so "a" and "b" are appropriate. Now here is the magic of POM. Those cells with "ab" must contain that number - we have found solutions. Those cells with no "a" or "b" (marked with a dash) cannot contain a 3.

The solver will return one of two types of elimination sets, which it calls "Rule 1" and "Rule 2"

Rule 1 considers each number in isolation. When looking for all the possible patterns for X it is possible that X may not appear in any pattern at all. If found, the solver reports and quits.

Rule 2 looks at all the patterns for all numbers 1 to 9. Within in each number all patterns may want to occupy certain cells - like a bottleneck. If that is the case then those cells are not available for other patterns used by other numbers. This is more cumbersome for a human to calculate, admittedly, but it works very well for the solver and we get a lot of this type. Patterns are pruned down and then Rule 1 is executed to find the first X where cells not used by X. Only the first X is reported - there may be other eliminations from numbers higher than X but it would be too confusing to report the total overlap.

The logic of POM ensure that there will always be at least one pattern for every X despite all these operations - unless the puzzle itself is faulty.


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Comments Talk

Thursday 23-Aug-2018

... by: dontsaymyid

OK, I found an SudokuX that uses Pattern Overlay Method.

Bowman's Bingo required 3 times : E5 is not 2, J1 is not 3, and D4 is 4.
POM will appear after first bingo.
Andrew Stuart writes:


Thursday 24-Aug-2017

... by: Benjedi420vt

Okay I just found an Xwing in the yellow pattern, sorry for being stupid. It's just hard to see these kind of things when looking at a piece of paper and doing in that way.


Thursday 24-Aug-2017

... by: Benjedi420vt

I'm a bit confused about how to apply this. In the side by side comparison of the two patterns...I can see how to figure out the purple pattern. If r2c6 is a 3, then the rest of the pattern does fall into place. However if the other candidate, r2c4 is a 3, the rest of that pattern DOES NOT FALL INTO PLACE.

In order for this strategy to be used by a human being and not involve guessing, you need to be able to find both patterns which I cannot figure out how to do in this case. A computer can quickly guess at a couple of the sticky points until it finds a fault and then use the other value as true to find all the patterns...but at that point it's just guessing anyway so why not just use a brute math calculation.

Maybe I'm just missing an easy to see reduction but in the case of a tutorial I believe it should be easy to see and pointed out so all reasonable users can see.



Sunday 17-Jul-2016

... by: David Spector

Sorry if this is off-topic, but it occurs on this page. You ask that commenters enter "letters you see". This makes the tragic assumption that you will never have a blind commenter (there is no way to "see" the CAPTCHA because it is an image). Blind people deserve to have the same good access to textual information that others have, just by virtue of being human beings. They should not be required to work extra hard to find a way around CAPTCHAs in order to post comments.

PS: I didn't enter the code correctly, even though I am sighted. Probably I have to push the Caps key?

I apologize that my comments are not relevant to the article. This is frustrating for me.
Andrew Stuart writes:

Sorry for the late reply, just now catching up on feedbacks.
I really dislike CAPTCHA but I had to add it to prevent robotic spamming. It was quite a problem. This one is home grown so I don't have a speaker feature. I can see that will be a problem for blind or partially blind. I'll see what I can find on the internet to replace it. BTW, the code entry field auto-captializes.

Tuesday 3-Feb-2015

... by: S.H.Khatami

I am looking the reference of Myth jellies in Pattern overlaying method. Could you send me this, please. I am writing a book and need that reference.

Friday 10-Jan-2014

... by: ralph maier

Actually at all the points where the 2 patterns overlap G2 J8 and E6 3 is the solution

Tuesday 14-Feb-2012

... by: Anton Delprado

I know it is always a fine line, particularly for the complex strategies, but this really seams like a trial and error strategy to me.

Also in the example G2 is a hidden single in Row G.
Article created on 12-April-2008. Views: 109321
This page was last modified on 9-January-2014.
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