Standard Proof for April 3, 2007

The following is an illustrated proof for the Tough Sudoku of April 3, 2007.

You may wish to refer to previous blog pages to properly understand this proof. Links to these pages are found to the right, under Sudoku Techniques.

This page illustrates only the deduced elimination steps that I choose to solve the puzzle. Although many other steps are possible, they are not shown, as to show every possible step is prohibitive in both time and space!

The information on the following blog pages may be helpful:

The illustrations of forbidding chains used in this proof will share the same 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
Strong and weak need not be mutually exclusive properties.

Puzzle at start

Puzzle start

A few Unique Possibilities are available here.

They are included in the next illustration.

Naked Pair 14

Naked Pair 14

Above, the naked pair 14 is illustrated as a continuous nice loop, or wrap-around, forbidding chain. One more cell is thus solved.

Hidden Triple 356

Hidden Triple 356

the hidden triple illustrated above is not easy to find. One can also use the naked quad formed by candidates 1479 at cells g9, h789 to make the same eliminations. After making the eliminations, i2=9% column. (Hidden single, as only one location remains for 9 in column i).

Locked Candidate 7

Locked 7s

Illustrated above as a forbidding chain, or AIC, a few 7s are removed from box h5. There are many additional such steps possible here, including some coloring eliminations with candidate 8. I struggled a bit here, finding many possible chains. Most of them were fairly deep. I am not sure why the following two chains did not immediately jump out at me, but they are very similar, and as such, not hard to find.

Depth 4 forbidding chain using candidates 1378

First depth 4 chain

Illustrated above is a standard short forbidding chain, heavily indicated by the standard puzzle marks. One could write this chain as

  • i1=1 == i1=8 -- i5=8 == i5=3 -- a5=3 == a4=3 -- a4=7 == i4=7 => i4≠1
The next chain uses three of the same strong inferences.

Depth 4 chain using canidates 138

Second depth 4 chain

Illustrated above a very similar forbidding chain:

  • i1=1 == i1=8 -- i5=8 == i5=3 -- a5=3 == a4=3 -- a4=1 == gh4=1 => i6≠1
After making this elimination, i1=1% column, followed by a cascade of Unique Possibilities until 41 cells are solved (UP 41).

Naked Triple 256 followed by coloring with candidate 6

Two steps

Highlit above, the naked triple 256 at d247 justifies the eliminations shown in column d. After making these eliminations, a short forbidding chain with candidate 6 is available:

  • b7 == i7 -- i6 == f6 -- f3 == d2 => b2≠6
The puzzle is now reduced to a cascade of naked singles to the end.


  1. Start at 22 filled - the given puzzle. Unique Possibilities to 27 filled. (UP 27)
  2. Pair 14 at de5 forbids afi5,de4,ef6=1 and forbids f5=4 UP 28
  3. Hidden triple 356 at g8,i89 forbids g8=1, i89=19 UP 29
    1. Locked 7s at i46 forbidsg4 h46=7
    2. i4=7 == a4=7 -- a4=3 == a5=3 -- i5=3 == i5=8 -- i1=8 == i1=1 forbids i4=1
    3. i1=1 == i1=8 -- i5=8 == i5=3 -- a5=3 == a4=3 -- a4=1 == gh4=1 forbids i6=1 UP 41
    1. triple 256 at d247 forbids d1=5,d3=26,d9=5
    2. fc on 6's b7 == i7 -- i6 == f6 -- f3 == d2 forbids b2=6 UP 81
  • Sets: 2+3+1+4+4+3+3 = 20
  • Max depth 4 at steps 4.2 & 4.3
  • Rating: .01 + .03 + 3(.07) + 2(.15) = .55 suitably tough


Time constraints will likely make blog pages more rare for the near future. Please feel free to let me know if a particular puzzle is very interesting, as I may not have time to look at each day's puzzle for the next few days - but I would hate to miss a very interersting one!!

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