Tough of January 30,2008 Page Three

The following is page three of an illustrated proof for the tough puzzle of 01/30/08. Page two and page one attempted to finely detail how to find eliminations. This page will continue in similar fashion. This page, and perhaps this entire proof, has a theme: remembering what one has found. A second related, but not quite equivalent, theme is: fully noting possibility matrix changes. Hopefully, this page will serve as a sterling example of how remembering works for me.

Previous blog pages may be helpful. Links to these pages are found to the right, under Sudoku Techniques. I may later list specific pages.

The illustrations of steps shown on this page will share this key:

  • 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
  • boxes highlight some groups
Please be aware that, for me, strong and weak need not be mutually exclusive properties.


After each elimination, including cell solutions, I find it helpful to manually update my puzzle mark-up. This forces me to be aware of what has changed. Thus, newly stronger strong inference sets are noted. Often, I spend some time examining those newly stronger (meaning more restricted) sis.

Puzzle Mark-up revisited

the mark-up above includes the result of step 3a: Locked 2s de7 => d9 ≠2. Note that candidate 8 required multiple updates. (8)ac9 both became circled. (8)e4 received an underline because of (8)e4=(8)ab4. The next step is the immediate result of those updates.

Steps 3a & 3b

Below, the Locked 2's elimination is graphed. Also, two different ways to achieve the same eliminations with candidate 8 are shown.

Locked 2's and coloring with candidate 8

Using only candidate 8, both of the following justify a5,d56≠8:

  1. b456=b1-f1=f56-e4=ab4
  2. b4=[row swordfish ace2, ae4, ac9] - a finned swordfish.
The latter requires both stronger 8s in row 9 and row 4. The one I found first required only the stronger 8s in row 4. Thus, although the finned swordfish is more pattern-like, the simple chain listed first is in my mind easier to spot given:
  • a single candidate elimination search has been previously conducted.
  • the underlined 8's in my puzzle mark-up at b1,f1,e4
It has been argued by others that finding such groupings is beyond typical human pattern recognition. This is not only silly, but clearly only true if one chooses not to ever look for them.

Finally, note the exquisite symmetry between the two chains making the same elimination: both have the same appendage that prevents a continuous, or wrap around situation: (8)b4. Furthermore, without (8)b4, both would have precisely the same targets. Somewhere, lurking within this symmetrical relationship lies a deep uniqueness of solution argument. But, I digress.

Step 3c

Below, the new Bivalue (1=2)e7 plays an important role. Some may prefer two chains.

Semi-remote pair like

The key to this one is the ALS (127)hi2. Thus, in that box we have the equivalent of the additional sis: (1=7)hi2 and (1=2)hi2. (1=2)hi2 is the most important in this deduction, as the new (1=2) at e7 forms the equivalent of a Y wing style:

  • (1=2)e7-(2)e3=(2)i3-(2=1)hi2
The chain, as shown above, also recognizes the continuation: (1)g1=(1)g5. Thus, the entire chain written more formally:
  • (1=2)e7-(2)e3=(2)i3-(pair27)hi2=(1)h2-(1)g1=(1)g5 => e25≠1
Thus, the key to this one, for me, is not only recognizing the sis (1=2)hi2 created by the ALS(127)hi2, but more importantly realising that the new sis, (1=2)e7, and the old sis, (1=2)hi2, can be easily linked.

Step 3d Prelude 1

Below, step 3c revealed an Almost Wrap Around Chain that is readily apparent if one looks for such animals. Moreover, note the overlap with the previous step.

Almost wrap-around chain

Finding Almost Wrap-Around Chains is usually worth the effort. This is because such an almost chain has mulitple weak links emanating from it, or alternatively, almost proves multiple sis. Recall that with a standard wrap around chain all the weak links are proven strong. Thus, above,

  • (1)f6=[wrap around chain: (1=2)e7-(2)e3=(2)i3-(pair27)hi2=(1)h2-(1)h6=(1)e6]
  • => (1)f6=sis list[(1)e67, (2)e37, (7)hi2, (2)i3hi2, (1)h26]
That is a lot of potential bang for the buck. In my head, I recognize (7)gi1=(7)c1 because (7)c1 is underlined in the puzzle mark-up. Thus, to remember this chain snippet, I note in the margin of my puzzle mark-up only: (1)f6=(7)c1. Since step 3c used four of the same sis, this shorthand is quite sufficient.

Step 3d Prelude 2


The ALS(23678)b456c6 creates the strong link:(7)c6=(8)b456. Although it proves other sis, this one creates a clear partition of candidates (78) in columns bc. Since I tend to prefer Hidden sets, what I saw was the equivalent condition using Almost Hidden Pair 49 at a45,c5. Step 3b eliminated (8)c6, and caused me to note this new stronger partition. Note therefor the Almost Chain:

  • [(8)c1=*(8)b1-(8)b456=(7)c6]-(7)c1
  • => (8)f1=[(8)c1=*(8)b1-(8)b456=(7)c6]-(7)c1=(7)gi1
which proves the sis:
  • (8)f1=(7)gi1
One can now trivially combine Prelude 1 with Prelude 2:
  • (1)f6=(7)c1-(7)gi1=(8)f1
  • => f6≠8 & f1≠1
Digression: After finding this chain, I noted that the elimination of (8) from c6 is not required to eliminate (8) from f6. Proving this is easy: (8)c6=[(8)f1=(1)f6]. Speculations about these sort of possible chains created the impetus for examining Almost Alternating Inference Chains (AAIC) .End of Historical Note Digression.

Step 3d - Linked AAIC

Below, I graph Prelude 1 and Prelude 2 together. The deduced chain, therefor:

  • (1)*f6=[(1)*e6=(1)*h6-(1)h2=(pair27)hi2-(2)i3=(2)e3-(2=1)e7]-(7)gi1
  • =(7)c1-[[(7)c6&(-8)**c1]=(ht479)a45c5-(8)a4=(8)b465-(8)**b1]=(8)**f1
  • => (1)f6=(8)f1 => f1<>1, f6<>8 native depth 11
For reference, and to illustrate how a Mixed Block Matrix handles such an animal, one can study the matrix below. Note that there are two Symmetric Pigeonhole Matrix blocks, both of which are indicated by bold characters. Please remember: the finding of such an animal can be done in at least two distinct, but formally related manners:
  1. Linking AIC chain snippets, as fully illustrated above
  2. Counting potential matrix rows versus columns, as described in previous blog pages
1r6 f6 e6 h6
e7 12
2r3 e3 i3
i2 27
h2 1 27
7r1 gi1c1
c6 72 6
b6 2 638
b5 2 638
b4 38
8r1 f1 c1 b1

Almost wrap-around chains linked

Not shown: Afer making these eliminations,

  • Step 3e: (Hidden Pair 58)f15 => f1≠9 & f5≠1369
This requires a refresh of my puzzle mark-up, as now (9)d1=(9)i1, (6)d5=(6)f6, (8)f1=(8)f5. Circling the (6) at f6 and (6) at d5 provided an essential clue for the next cascaded step.

Step 3f Prelude


  • (6)d5=[(6)f6-(1)f6=(7)c1see previous step]-(67)c6=(2)c6
  • => (6)d5=(2)c6.

Using what I already know

At this point, I note in my puzzle mark-up both 1 & 2 underlined at b8. These serve as a clue for a helpful continuation to (6)d5=(2)c6.

Step 3f


Above, to make the graphing easier, I have replaced (1)b8=(1)b13 with (1)b8=(1)a78. Also, (2)b8=(2)b56 is replaced with (2)c89=(2)b8. However, the underlined (12)b8 meant the latter. The chain now reads:

  • (6)d5=(2)c6-(2)c89=(2-1)b8=(1)a78-(1)a2= [(1) d2=h2-g1=g5]
  • => d5≠1
  • => circle (1)e6,(1)f6 => (not shown)
  • step 3g Locked 1s ef6 => h6≠1
If one cares to, one can find a chain, depth 9, that replicates step 3f. I generally do this, as it helps me to understand other things about the puzzle. I guess one could call that activity puzzle study. I find it often helpful. Here, my notes say D9 poss, meaning independent Depth 9 chain possible to achieve the same elimination. Independent means that it requires no knowledge of any previous steps. But, again I digress!

For reference and illustration purposes, below find an 8x8 Triangular Matrix that is actually depth 9. In bold, note that I treat the ALS(127)hi2 as a single Boolean:(1)h2=(pair27)hi2. This can be done freely without any loss of generality.

1g g5 g1
1r2 d2h2a2
1b b13b8
2b b8 b56
hi2 1 p27
7r1 gi1c1
c6 276
6Bx5 d5 f6

Alternate Step 3f - simple TM

This content has been added later. First, let me apologize. Oftentimes, I find many ways to justify the same elimination. What follows is another way to achieve step 3f. One could write this alternative as an AIC, but it is a bit messy. The idea has some overlap with the previous two manners shown to achieve d5≠1:

  • [(1):g5=g1-h2=*d2]=*(1)a2-[(1)b13=(1-2)b8=(2)b56-(2=7)**c6-(7)a45=(7)a2]=(6)**c6-(6)f6=(6)d5
  • => sis[(1)g5,(1)d2,(6)d5] => d5≠1
The key to finding this one is the confluence of underlines:
  • (1)b8, (2)b8, (7)a2
  • yielding [(1)d2=(1)h2]=(6)c6 when combined with (267)c6
  • As a TM, depth 7:

    1g g5 g1
    1r2 d2h2a2
    1b b13b8
    2b b8 b56
    7a a2 a45
    c6 276
    6B5 d5 f6

    Step 3h - ALS again.

    ALS & AHS

    The ALS(127)h26 became a bit stronger with the introduction of the bivalue (2=7)h6. Having noted earlier the relationship:

    • (2)g789=(2-1)g5=(1)g1 in step 2c
    It is easy now to find:
    • (2)g89=(2-1)g5=(1)g1-(1)h2=(pair27)h26
    • => h9≠2
    • => (4)h9 %cell, thus UP 28
    After this step, I noted but did not use:
    • AUR 27 hi26 => (38)i6=[pair 27 @ h6,(2=7)h45] => i6≠27
    justified by mxing and matching naked/hidden sets:
    • (naked pair27)i26-AUR(Hidden pair27)h26=(pair27)h6,h45
    Although this elimination is in my scrap heap, I thought some may find it interesting in format. This step becomes available after (4)h9 because of the newly stronger 2's column h. This new strength is indicated by the underline one would place under 2 at h2, meaning (2)h2=(2)h456.

    This proof continues on Page four. Significant progress has been made.

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