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.

### Starting Point

### 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:

- Cells limited to two candidates
- 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

- All the cells with more than two circles in them
- 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

- All the cells such that
- There are two circles
- They
*see* a two candidate cell

- All the remaining cells with two circles
- All the cells with two candidates
- All the one circle cells.

Each iteration above has an internal hierarchy:

- Start with the candidate that is circled most often
- 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 =*- strong link endpoints
*Black lines =*- strong links
*Red lines =*- weak links
*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:

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

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