Diabolically Tough Puzzle of February 6, 2007

The following is an illustrated proof for the Tough Sudoku of February 6, 2007. Since this is a very difficult puzzle, I may have easily missed something(s). Certainly, there are multiple ways to tackle this one. If you are interested in a logical approach to solving truly difficult puzzles, this page is for you. If, however, quick solving is your goal, then this page is probably a waste of your time.

Since this proof uses a Y wing style, some forbidding chains, (also called Alternating Inference Chains) and a variety of other techniques, you may need to refer to previous blog pages to understand this proof. Links to these pages are found to the right, under Sudoku Techniques.

At many times during this illustration, there are other steps available. It is not the goal of this page to show every possible step, but rather to illustrate steps that, taken together, unlock this puzzle

Puzzle at start

PUzzle start

A few Unique Possibilities:

  • f9 = 6% box & row
  • h7 = 3% box & row
  • i5 = 3% box & column
  • b1 = 3% row
  • g7 = 1% box
Unique Possibilities get the puzzle to 28 cells solved. (UP 28)

Hidden Pairs 58 and 15

Hidden pairs

As if often the case, some Hidden pairs can be located before considering the possibility matrix. Illustrated to the left are

  • Hidden Pair 58 at ab9:
    • forces ab9=58
    • forbids ab8=5
    • forbids a7=8
    • forces a7 = 2% cell
  • Hidden Pair 15 at ah2
    • forces ah2=15
The puzzle is now at 29 cells solved. (UP 29)

Y wing style wrap around chain - also called Hub and Spoke configuration

Hub and Spoke or Y wing style wrap around chain

This type of Y wing style chain happens so frequently, it is probably a good idea to search for it all the time. It is much more common than a standard Y wing. As a wrap around chain, it is more powerful. Unfortunately, it usually involves considering a grouped argument, such as the 1's in box b2.

  • a8=1 == a8=9 -- a1=9 == c1=9 -- c1=1 == a12=1
    • forbids a5=19
    • forbids c1=26

In conjunction with the chain above, the following additional eliminations are justified:

  • Locked 9's at ac1 forbids 9 from the rest of row 1
  • Newly locked 2's at bc3 forbids 2 from the rest of row 3

Forbidding Chain using four strong sets

First depth 4 chain

Illustrated above is:

  • g5=5 == g3=5 -- b3=5 == b9=5 -- a9=5 == a9=8 -- a5=8 == a5=6
    • forbids g5=6
  • g4 = 6% box
  • h4 = 7% box
  • Puzzle is now advanced to 31 cells solved (UP 31)

Y wing style - very common type

Very Common Y wing style

The newly revealed 24's at g1, h9 are a common key to look for this type of Y wing style elimination pattern:

  • g1=4 == g1=2 -- g5=2 == h56=2 -- h9=2 == h9=4
    • forbids h1=4
    • forbids g9=4
  • forcing:
    • g1 = 4% box
    • h9 = 4% box
  • advancing the puzzle to 33 cells solved (UP 33)

To this point, the puzzle eliminations have been indicated by the search parameters previously explained in Forbidding Chains 102 The Practice, or in the Y wing styles page. The following chains are also indicated, but now there are dozens of possible paths. Most of the paths indicated will eventually lead to a solution. What follows here is the least deep - meaning least maximum number of native strong sets per step - path that I found. Probably, there are better paths. Also, you may well prefer fewer chains of greater depth.

Forbidding Chain using five strong sets

Depth 5 forbidding chain

Illustrated above is the following:

  • g5=2 == g5=5 -- g3=5 == b3=5 -- a2=5 == a9=5 -- a9=8 == a5=8 -- h5=8 == h6=8
    • forbids h6=2
  • Newly Locked 2's at gh5
    • forbids bcdf5=2
  • Old locked 8's at h56
    • forbids h1=8

Y wing style

Y wing style

The Y wing style shown above has been available for a while. Grouped candidates such as the grouped 2's at def2 are a common theme with Y wing styles.

  • i1=8 == d1=8 -- d1=2 == def2=2 -- i2=2 == i2=6
    • forbids i1=6

Y wing style - Again

Y wing style again

This Y wing style has also been available for a while. It is indicated by the standard puzzle mark-up, but is the type I generally have the most difficult time spying.

  • c3=6 == a1=6 -- a1=9 == c1=9 -- c1=1 == c5=1
    • forbids c5=6

X wing on 6's

X wing

The last two Y wing style eliminations have exposed this pattern:

  • X wing on 6's at ad15
    • forbids d26=6

Depth 4 forbidding chain with multiple effective endpoints

Depth 4 AIC

The newly revealed bi-value strong 6's in row 6 indicate the existence of this chain:

  • f6=3 == e6=3 -- e6=6 == c6=6 -- a5=6 == a5=8 -- h5=8 == h6=8
    • forbids f6=8
    • forbids e6=8
In order to see that e6≠8, consider that the chain also indicates:
  • e6=6 == h6=8

Forbidding Chain using four strong sets but only two candidates

Two candidate Forbidding Chain

The Forbidding Chain above has been available for quite some time. Basically, the sevens act as a bridge for the fives. In this way, it is very similar to a single candidate coloring elimination.

  • f8=5 == d8=5 -- d8=7 == i8=7 -- i3=7 == g3=7 -- g3=5 == g5=5
    • forbids f5=5
f5 is starting to look like an important cell. Oftentimes, one cell becomes a focal point for unlocking a puzzle.

Another four strong set forbidding chain looks at cell f5

Another Depth 4 forbidding chain

Again, this chain has been hanging around waiting to be used

  • f3=8 == f3=3 -- f6=3 == e6=3 -- e6=6 == c6=6 -- a5=6 == a5=8
    • forbids f5=8
Since we have whittled that cell f5 down to almost the minimum, perhaps it is useful now in a chain.

Forbidding Chain using the newly uncovered bi-value strong set

Depth four chain using new strength at f5

The newly revealed 14 at f5 is useful

  • f2=2 == f2=4 -- f5=4 == f5=1 -- c5=1 == c1=1 -- h1=1 == h1=2
    • forbids d1=2
    • forbids i2=2
  • forcing i2 = 6% cell
  • advancing the puzzle to 34 cells solved (UP 34)
That was much work just to get another cell solved. But, it has been a set up for one final move. There are many equivalent moves available at this point. I choose to use the newly strong relationship between 68 in cell d1, but there are many other paths to take here.

Forbidding Chain using five strong sets

Depth 5 chain

  • d1=6 == d1=8 -- f3=8 == f4=8 -- f4=1 == f5=1 -- c5=1 == c1=1 -- c1=9 == a1=9
    • forbids a1=6
  • forcing:
    • d1 = 6% row
    • i1 = 8% row
    • % in some container to the end
  • UP 81

Solved Puzzle


Summary data

Sets: 2 + 2 + 3 + 1 + 4 + 3 + 5 + 1 + 3 + 3 + 2 + 4 + 4 + 4 + 4 + 5 = 2(1)+ 3(2) + 3(3) + 5(4) + 2(5) = 47 YIKES!!!

Max depth 5, twice - so not very deep

Rating: 2(.01) + 3(.03) + 3(.07) + 5(.15) + 2(.31) = 1.69 - so fairly monstrous

Indicate which comments you would like to be able to see

I maybe scarce for a couple of days. I intend to produce a blog page proof for the very, very difficult tough puzzle of November 10 2006. It is likely that the proof will consume two blog pages. Also, the proof will involve mutliple complex advanced forbidding chains.

The cold weather has More...
That all looks great, but I have absolutely no idea what any of it means :D Is there any way at all it can be simplified for those of us without a degree in mathematics?
Hi Steve. There seems to be no bounds to what fcs can achieve. Have been reading up on Nice Loops (in Paulspages, with some nice examples if youll forgive the pun)- and they appear to be just wraparound fcs. Is the only difference in nomenclature? A question - do you familiarise yourself with the More...
Hi Bebe!
I have taken an approach that is mathematical, or logical, in nature. Primarily, the mathematics serves as a sound foundation for the techniques.
I am musing over how to simplify the ideas without compromising them.
Hi giblet!
There are many different names for the same ideas throughout the web. Nice Loops is far more standard. I intend to update previous blog pages to reflect some of the more standard terminologies. 'wraparound' was coined at this site, (perhaps by me), when I uncovered the idea without More...
I did 11/10/06. Rough one. No fancy way of how. did the naked quads, the pointing pairs. nothing left but to color my way through. Good challenge. I often go through the archives. Night man. 12:52
Upon careful review of the proof I have developed for the tough of 11/10/06 - I have decided to not blog that proof just yet. Instead, I probably will post a blog page reflecting more information about advanced forbidding chains.
Hey Steve! I now see there's an easy proof blog for a tough puzzle! I've bookmarked it to peruse at leisure, and thanks so much for taking the trouble. :D
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