Strategies for Popular Number Puzzles

Death Blossom

(a.k.a. Aligned ALS Exclusion)

This strategy is based on extending Aligned Pair Exclusion but uses Almost Locked Sets to make some clever reductions. From the components used it could be named Aligned ALS Exclusion but Mike Barker, who formulated it first in this thread, hit on "Death Blossom" because it starts with a cell designated as the "stem" which points to Almost Locked Sets, or the "petals", and is a great deal more flowery.

An Almost Locked Set is any group of N cells (that can all see each other) with N+1 candidates between them . This includes bi-value cells. A Locked Set, by contrast, contains exactly the right number of candidates for the group, examples of which are Naked Pairs and Triples.

Death Blossom 1
Death Blossom 1 : From the Start
To get a feel for what’s going it worth working backwards from the elimination of 7 in C3 in this first example. We have two Almost Locked Sets {A3,E3,F3} and {C5,C6,C8} and they both have 7 as a common number between them. If C3 did have 7 as a solution it would reduce the first Almost Locked Set to a 5 on F3, 1 on E3 and lastly 3 on A3. The second ALS would reduce to a Naked Pair of 6/8 forcing C5 to be 1. If A3 is 3 and C5 is 1 then the stem cell (coloured green) A5 is left with nothing. This confirms that C3 is not 7.

Reversing direction will help illuminate the Death Blossom idea since we start with green “stem” cell. A5 with {1/3} must be able to see at least two ALSs which contain all its candidates. It is important that the 3 in A5 can see all of the 3’s in the yellow coloured ALS and the 1 in A5 can see all of the 1’s in the brown coloured ALS (one instance each in this case). But A5 overall does not have to see every single cell in all the ALSs, just the cells it shares candidates with.
Now, the two ALSs must have a candidate Z in common which the "stem" cell does not have. Because ALSs contain exactly one extra candidate for the number of cells they occupy (the N+1 candidates for N cells rule), we can assert that ANY cell that can see all the Zs in both ALS but is not part of those ALSs or the stem cell can be removed. Such a cell is C3.

Death Blossom was discovered by extending Aligned Pair Exclusion (APE) and asking if there was generalisation beyond the pairs and triples discussed in Aligned Pair Exclusion. With Almost Locked Sets there is. The stem cell A5 and the elimination cell C3 can’t see each other – they not aligned, but the pairs they can make do affect the board. In our example, consider the pairs that can be made between the 7 in C3 and the 1/3 in A5. These are 1/7 and 3/7 in C3 and A5 respectively. Both these turn out to be illegal since they would reduce our ALSs to having less candidates than cells. So whatever the solutions to the two disparate cells C3 and A5, C3 will never contain a 7.

Death Blossom 2
Death Blossom 2 : Load Example
In Figure 2 quite a different arrangement is apparent but the logic is identical. The two yellow coloured cells form a two-cell ALS with the values {3/5/7}. At the bottom there is a four-cell ALS with {2/3/6/8/9}. Our stem cell again contains only two candidates {6/7} (coloured green) but don't think this is a restriction. There could be five or six numbers in the stem cell. As long as there are sufficient Almost Locked Sets that the candidates can see then the pattern can be made to work.

There is another way to look at this example which mirrors some strategies already covered. Starting in G3

If G3 is 7 -> D3=5 -> D7=3 therefore D9,E9,F9,J7 <> 3

If G3 is 6 -> G8=9 -> G7=2 -> G9=8 -> J9=3 therefore D9,E9,F9,J7 <> 3
Much of the first-sight complexity seems to evaporate when traced through in this manner.

Note: Example 2 cannot be found by the solver unless Digit, Cell and Unit Forcing Chains are turned off. I retain it as an example because it's the best one to show multiple eliminations. Indeed 4 eliminations is a pretty good hit. Unfortunately I can't find the orginal Sudoku to give a start position.

Massive DB
Massive DB : Load Example or : From the Start

Here is a fun one by Klaus: two massive ALSs in a DB puzzle in only 2 boxes.

Death Blossom
(ALS:{D3|E2|E3|F2|F3} and ALS:{E4|E5|E6|F4|F5}) + Stem:E1 means 7 can be removed from F1

Update August 2013:

Other fun examples Klaus Brenner has found include a Death Blossom with six eliminations and a Death Blossom with all its ALS in only one Box.

Go back to Sue-de-CoqContinue to Bowman's Bingo



... by: Ghost 2T

Tuesday 17-Jan-2023
All of the flowery variations above are in the form of ALS.XZ.

It was born just to mess up the brain and waste the player's time. (except for the "Death blossom" pattern I see appearing outside of this web, because it has an overlap of hidden Z)


Monday 7-Jun-2021
This one took me quite a while to understand. I suppose my easier way of understanding this is more of "Cell Forcing". Because you would literally "force" all candidates on the stem cell to make up one complete locked set on at least one of the "petals". And when I explain about sudoku strategies to my friends, I like to refer potential eliminations as candidate X. So if I'm right, my way of understanding this is: "If forcing all values in the stem cell can create at least one locked set from each 'petal' (more specifically ALS), then X can be eliminated from any square that can see all instances of X from all ALS."

I have yet to find one though, even though it's a nice logic and want to get one so badly... The best I've ever found is ALS, and whenever I use your solver, Aligned Pair Exclusion always picks it up first, haha.

... by: Joe Bleau

Sunday 26-Jan-2014

You wrote :
“Klaus Brenner has found a Death Blossom with five eliminations”

If I’m right, there are 6 elimitaions.

Here are the 5 found by your program :
Death Blossom
([ALS:F6] and [ALS:E6|E7|E9]) + Stem:F7 means 4 can be removed from B6
([ALS:F6] and [ALS:E6|E7|E9]) + Stem:F7 means 4 can be removed from D4
([ALS:F6] and [ALS:E6|E7|E9]) + Stem:F7 means 4 can be removed from D5
([ALS:F6] and [ALS:E6|E7|E9]) + Stem:F7 means 4 can be removed from E4
([ALS:F6] and [ALS:E6|E7|E9]) + Stem:F7 means 4 can be removed from G6

I think we can add this one :
([ALS:F6] and [ALS:E6|E7|E9]) + Stem:F7 means 4 can be removed from F4
Andrew Stuart writes:
Hi Joe
I have a rule in the solver that says any elimination cell can't see the stem cell. But reviewing the example and definitions I am wondering if this is a necessary rule. Removing it I get your 6th. It will take some time to test for a negative as DBs are very rare, being at the end of the strategy list. But I'll give it a go

Edit: released a version with this rule relaxed, seems harmless to do so. Now finds six - [Del]
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... by: last1x {rus}

Monday 16-Apr-2012
I've read some of your articles (in "extreme strategies" section mainly) …good analysis! Nevertheless I take a risk to suggest that all strategies (even the "baby" ones) are the "chains" and could be described by "backtracking" programming method. Imagine the cell with one possible candidate. If you try to chain it you'll have no success with that ;) Let this be 1-dimension chain (starts only with A). Go next: 2 candidates, => 2 dimensions (we can start a chain with A | B). And so on… Chains can be opened (XYZ Wing) or closed (Naked Pair). Anyway you can build your own chains to perform eliminations. Higher the dimension and larger the length of chain then more Brute-Force strategy resembles. But I like the name “Death Blossom” too ;)

... by: Thinkist

Sunday 25-Mar-2012
It looks like AICs with ALSs and forcing chains accomplish the same thing, making this technique obsolete. I ran both of your examples through the solver, and the Death Blossom was not found.
Andrew Stuart writes:
Almost true. These are fairly old examples and I need to replace them. Last year I improved the AICs used in the Forcing Chains to include some additional starting positions and DB has now become extremely rare. I have four examples from the 300k stock I ran to update the The Relative Incidence of Sudoku Strategies article, so it's not completely obsolete, but then again, there maybe something subtle missing from the AICs with ALSs that I don't have. I have a soft spot for DB as a piece of outrageous and elegant pattern based logic.

Update: 25 March 2012. I've replaced the first example with one of those four. - [Del]
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... by: Bud Nienhaus

Friday 30-Apr-2010
I don"t see why these examples couldn't simply be called ALS-XY-wings, rather than death blossoms.
Andrew Stuart writes:
How could I not use the name coined, which is so much more evocative :) - [Del]
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... by: STRMCKR

Friday 21-Aug-2009
Reversing direction will help illuminate the Death Blossom idea since we start with green “stem” cell. G8 with {3/5} must be able to see at least two ALSs which contain all its candidates. It is important that the 3 in G9

THERE IS A TYPO in the first reference cell G8.
it should read G9.
Andrew Stuart writes:
Well spotted. Fixed - [Del]
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