# Forbidding Chains 102 The Practice - Sudoku Solving Technique

Many places with tips, tricks, techniques or help on how to solve sudoku puzzles completely forgo a complete discussion of forbidding chains. Probably, this is because finding the chains, that is the real trick.

To gain a bit more traction,

Open the trunk, target the chain.

Espy them with some passion,

As to idle same, quite a bane.

Tritely and often, language is avowed as less succinct than images. In deference to this adage, this page will amply employ the latter.

## Complete Puzzle Proof

The following is a complete proof of the Tough puzzle of January 15, 2007.

### Possible locations for 4

Because of the three original 4's, clearly ab3,c2,ab1 ≠ 4. Thus one could write:

• b2 = 4%Box. Meaning: b2 = 4 because it is the only 4 possible in that box.
Some may choose to somehow optimize their search for UP's at the beginning. I basically just scroll through the candidates, unless I see something obvious.

### Possible locations for 5

Above we have a good example of eliminations that can be made during set up. Note the strong 5's at c46 and g46. One could immediately eliminate 5 from b456 and h456. If one does that, now we also have strong 5's at df5. One can additionally eliminate 5 from e4,ef6. One could instead make all these eliminations in one step:

• c4=5 == c6=5 -- g6=5 == g4=5 forbids b456,h456,e4,ef6=5
• This is really just an X wing, and the eliminations are justified by
• c4=5 == c6=5 forbids b456=5
• g6=5 == g4=5 forbids h456=5
• c4=5 == g4=5 forbids e4=5
• since g4=5 -- c4=5, the additional wrap around chain result
• c6=5 == g6=5 forbids ef6=5

### Some more Unique Possibilities (UP)

To avoid the tedium of analyzing each indiviual candidate in this manner, I trust you can ascertain the validity of the following cell solutions:

• a8 = 7 %Box
• h8 = 8 %Row
• h7 = 3 %Row and %Box - note we need the previous UP to get this one
• g2 = 1 %Cell - again, need the previous UP

### More eliminations at UP 28

Before filling in the possibility matrix, a few more eliminations are possible. Note above the following:

• Locked 2's at g46 forbids i456=2
• Locked 8's at ab1 forbids ef1=8

### Possibility matrix at 28 filled

If you have not erred, and made the eliminations noted above, your current possibility matrix should look exactly like the one above. Here, none of the following techniques will yield any eliminations: subsets (naked or hidden), locked candidates, coloring, xwings, swordfish, Y wings. There is, however, at least one easy to spot forbidding chain.

How does one spot forbidding chains? The method that I use is progressive. I start looking where I believe chains are most likely to exist, and progress towards the least likely. Another factor in the search order is the efficient advancement of the puzzle.

Since forbidding chains use strong sets as their primary building block, it is logical to look at the native strong sets first. I begin by identifying the strongest of the native strong sets. Of these, there are two types:

1. Cells limited to two candidates
2. Candidates limited to two locations in a large container
Generally, I print out the puzzle and mark it up. Since if the puzzle is in an early stage, (less than about 35 cells solved), there usually are more of item 2, my focus for this puzzle will be primarily on sets strong by location. Here is a typical attack:
• First, make sure I did not miss any coloring eliminations. Since coloring involves only one candidate at a time, it is easy to spot. Also, the Coloring search gives me a feel for the puzzle.
• I print out the puzzle. It is time to find a pen (I despise pencils) .
• All the candidates limited to two locations within a large container get circled
• All the cells limited to two candidates get committed to memory - you may choose to mark them

Search plan, from most important to least

1. All the cells with more than two circles in them
2. All the cells such that
• There are two circles
• There are only two candidates
• Are not part of a pair
• Why not these? Most of their strength is spent
3. All the cells such that
• There are two circles
• They see a two candidate cell
4. All the remaining cells with two circles
5. All the cells with two candidates
6. All the one circle cells.
Each iteration above has an internal hierarchy:
• Progress towards the candidate circled least often

### First puzzle mark-up at 28 filled

With this puzzle, there are no cells with more than two circles in them, so the search defaults to item 2. The most promising start point then is cell f7 = 25, with two circles. Here, I quickly find a short chain.

### Forbidding Chain Found

When I first started to dabble in forbidding chains, I would diagram them on the puzzle much like the image below. This helped me not only to visualize what was going on, but also to check the chain for validity.

Key:

Black circles =
Black lines =
Red lines =
Green circles =
Eliminations
Notice:
Here is one forbidding chain representation of this step:
• f7=5 == f7=2 -- f1=2 == e1=2 -- e1=5 == e9=5 thus:
• f7=2 == f1=2 forbids f56=2
• e1=2 == e1=5 forbids e1=6
• f7=5 == e9=5 forbids d9=5
After making these eliminations, we have still no more Unique Possibilities. So examine the new puzzle.

### At 28 filled after wrap around chain

The partial puzzle above has:

• Naked triple 129 in the blue cells
• Hidden triple 568 in the yellow cells
Each of these forbid exactly the same things:
• abe9 = 1
• e9 = 2
• abe9 = 9
After performing these eliminations, there are some Unique Possibilities. Rather than illustrate all these, I will just list them:
• e8 = 1% Box & Column
• c6 = 1% Column
• c4 = 5% Box & Column
• g4 = 2% Cell
• g6 = 5% Cell
This gives us 33 cells solved.

### Puzzle at 33 filled

Here, the search begins all over again. There are a couple of locked sets eliminations possible here, but... while looking for Unique Possibilities, I noticed an easy chain with candidate 6.

• 6's in column c are limited to c2,c8
• 6's in box e8 are limited to e9,f8
• Conclude:
• c2=6 == c8=6 -- f8=6 == e9=6 forbids e2=6

Happily, this elimination unravels the puzzle! If needed, below are the Unique Possibilities to 81 filled cells. (UP 81).
• e2=8 e4=9 e6=2 e1=5 e9=6 a9=8 b9=5 f8=4 d8=9 d9=2 f7=5 c8=6 all %cell
• c2=9 f5=7 f6=8 d4=4 d5=5 b5=6 a4=3 b4=7 b6=9 a6=4 a5=2 b7=1 all %cell
• a7=9 a1=6 a3=1 b3=3 b1=8 c3=7 c2=3 f2=6 f1=2 i2=7 i3=6 i4=8 all %cell
• h4=6 i6=3 h6=7 h7=4 i7=2 i1=9 h1=3 h3=5 h5=1 i5=4 i9=1 h9=9 all %cell

### Solved Puzzle

Complete proof of this puzzle in the style I usually use:

1. Start at 23 filled - the given puzzle. Unique Possibilities to 28 filled. (UP 28).
1. Locked 4's at gh7 forbids f7=4
2. X wing on 5's at dg46 forbids bh456,de4,ef6=5
3. f7=5 == f7=2 -- f1=2 == e1=2 -- e1=5 == e9=5 forbids d9=5, e1=68, f56=2
4. triple {29,19,129} at (dgi9} forbids abe9=19,e9=2 UP 33
2. fc on 6's: c2 == c8 -- f8 == e9 forbids e2=6 UP 81
• sets: 1 + 2 + 3 + 3 + 2 = 11
• Max depth 3: at step 2.3 and step 2.4
• Rating: .01 + 2(.03) + 2(.07) = .21

Practice puzzles: I shall eventually add some specific ones. With the information contained in the blog up to this point, one can solve:

• All the Tough Puzzles at Sudoku.com.au from year 2005
• One can use the archive link at the top of the page to access these puzzles