Easter Monster Round 5

The following is the fifth page of an illustrated solution of the Easter Monster.

If this is your first visit to this blog, WELCOME!!

Previous blog pages may be helpful. Links to these pages are found to the right, under Sudoku Techniques. Specifically, it may be helpful to have visited the following pages:

• Opening Volley on the Easter Monster
• Matrices in Sudoku
• Round 2 with the Easter Monster
• Easter Monster Round 3
• Fourth Chronicle of the Easter Monster Battle
• Forbidding Chains 101 The Theory. Note that I now use the term Alternating Inference Chain (AIC) to mean Forbidding Chain.

The illustrations of steps shown in this proof will share the new key style:

• black line = strong inference performed upon a set (strong link)
• red line = weak inference performed upon a set (weak link)
• black containers define a partioning of a strong set(s)
• candidates crossed out in red = candidates proven false
• Orange labels mark derived inferences
• Blue circles indicate proven strong inference set result
• Green circles indicate intermediate strong inference points
• Brown numbers indicate approximate chain order
• Other marks provided prn

Please be aware that, for me, strong and weak need not be mutually exclusive properties.

Previous information

Previously, many derived inference groups were proven. This page will use the following sis groups:

• Each set in Group Z below contains exactly one truth:
  • (27)r2c13,(16)r2c56,(16)r2c79
  • (16)c8r13,(16)c8r45,(27)c8r45,(27)c8r79
  • (27)r8c79,(27)r8c45,(16)r8c45,(16)r8c13
• The result column sis from step 2g
  • (2):[r4c5,r6c7,r4c1]

Step 2j prelude using Kraken Cell Transport.

Since this puzzle resists most normal attempts to crack it, it seemed likely that some research might be required into relationships to locate good attacks. On the previous page, the guardian sis was used to locate a likely chain. Here, the relative symmetry of candidates 345 in relation to cells r7c9 and r9c7 caused me to create a transport list out of each cell.

Basically, the goal is take the sis inherent in a cell and find some equivalent sis at least partially out of the cell. Although I created many such lists, some of them seemed more likely to be fruitful than others. In this case, the transport list for r7c9 seemed more likely to be fruitful than the one for r9c7, primarily because candidate 67 interactions uncovered in the guardian step.

Above, I am illustrating three short transportation chains.

1. (349=5)r7c9-(5)r7c46=(5)r9c46
2. (459=3)r7c9-[(3):r79c8=r3c8-r3c4=r7c7]=sis(3):[r1c48,r9c4]
   • Note that there are many such chains possible, but I wanted one that interacted strongly with canidates 67.
3. (359=4)r7c9-[(4):[Xwing r79c25]=r1c2-r1c7=r3c9]=(4)r3c2
   • This one is nice, as it requires only a single value transported out

PUtting all three of these together we get:

• Old sis (3459)r7c9 =>
• sis[(35)r9c4,(5)r9c6,(3)r1c4,(3)r1c8,(4)r3c2,(9)r7c9]

This new sis seems a bit unwieldy, but a short look at some of the previous chains considered makes it very promising. The only item that cannot be resolved easily is that darn (9)r7c9. However, one may see a potential logjam primarily because of sis[9 column 3].

Step 2j (2)r2c1=(6)r6c1

Below, the transported portions of cell r7c9 are labelled with a green k. The rough order of chained sis are marked with brown numbers. Only strong inference sets are marked. The Z argument group is used twice, and labelled accordingly.

The easiest way to describe such a long chain is probably a Triangular Matrix

2B1

r2c1

r3c2

Z27

r65c2

7r6c2

7r1

c2

c4

7c6

r2

r9

6r6

c1

c2

c6

1r6

c2

c6

c3

1r8

c3

c45

6B8

r9c6

Zc8

r9c4

6r1

c4

c6

c8

2jK

*

*

*

*

*

9r7c9

r6c9

7

9

3

r2c9

6

3

8

r2c1

2

8

3

3B4

r6

c1

r5c3

9c3

r6

r7

r5

It is a bit unusual for a Triangular matrix of this length to have only two items listed in the first column. The result column reads:

• (2)r2c1=(6)r6c1
• =>r6c1≠2

This leads to a few relatively easy eliminations, although all of them use proven sis.

Step 2k as an immediate result of <2j> and <2g>

Below, recall step 2j. Proven in that step was sis(2)[r6c1,r7c6,r4c5]. However, we have just shown that (2)r6c1 is false, therefor we can update the strong inference set to:

• 2g 2:[r7c6,r4c5]

The logic for this step is fairly simple

• (2)r4c6=(2)r6c7 => r4c7≠2

Remembering what has been already proven certainly makes that one a breeze.

Step 2L as a Y Wing Style using 2j

Below, recall 2j sis [(2)r2c1=(6)r6c1]. Add a pair of native strong inferences.

The AIC for this one could be written:

• (1=6)r6c6-(6)r6c1=2j(2)r2c1-(2)r4c1=(2)r4c5 => r4c5≠1

Step 2m as a depth 4 chain using 2j and Z

Below, recall 2j sis [(2)r2c1=(6)r6c1]. Add one native strong inference and use strong inferences from the Z group twice.

The AIC for this one could be written:

• (1)r8c3=Z(6)r8c1-(6)r6c1=2j(2)r2c1-(2)r4c1=(2)r4c5-(2)r8c5=Z(27)r8c4
• => (1)r8c3=(2=7)r8c4 => r8c4≠1

After just a little practice, using derived strong inference sets seems just as natural as using native strong inference sets. Although the structure of a derived strong inference set may be a tad unconventional, the logic is precisely the same.

Digression

Some large and unconventional weaponry seems warranted with this particular puzzle. The practical use of most of these ideas will be very small. However, mini-cases of the same logic type are possible. In all but a very few puzzles, the depth and complexity found here will not be encountered.

I have a few more unconventional attacks to attempt. On the next page, I probably will employ an sis derived from an Almost Unique Rectangle - more aptly named, I think, a Forbidden Rectangle.

Thanks for your patience!