# Steps to Solve Tough Sudoku of January 21 2007 with Proof

The following is a graphical illustration of a proof for the Tough Sudoku of 01/21/07. 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.

### Puzzle at start A few Unique Possibilities:

• c8 = 8% row
• h5 = 8% row & box
• g8 = 5% row
• i8 = 1% row
• h1 = 5% box
Thus we have UP 27. This yields:

### Puzzle at UP 27 If I am filling in the possibility matrix by hand, I like to locate hidden pairs while doing so. In this puzzle, two such hidden pairs are available:

• 36 at gh7
• 29 at g12
Also forbidding:
• 6 from e7
• 3 from ghi9
• 29 from g46
• 2 from g79

### Possibility Matrix after Hidden Pair eliminations Now the following cells solve:

• i9 = 2% cell
• e7 = 7% row
• c3 = 7% row
• g3 = 8% box
• b1 = 8% box
• f2 = 8% box
• b7 = 2% column
• a7 = 4% cell
• c7 = 9% cell
• d2 = 7% box

The puzzle is thus advanced to 37 cells solved (UP37).

### PUzzle at UP 37 At this point, there are many eliminations available. It would be unwieldy to list them all. Instead, I will only illustrate the eliminations that my proof requires. Amongst the possible eliminations not listed are:

### Y wing style eliminations at UP 37 The forbidding chain illustrated above can be written as:

• e3=1 == d1=1 -- d1=6 == c1=6 -- c1=4 == b3=4
• forbids e3=4
• and forbids b3=1

This allows us to solve three more cells:

• e3 = 1% row
• b3 = 4% row
• d4 = 1% box & column

### Puzzle at 40 cells solved (UP 40) Again, there are many possible eliminations. One that my proof requires is illustrated here.

• Locked 4's at def5
• forbids e4, ef4=4

Another Y wing style elimination is available also, and is illustrated below.

### Y Wing Style elimination at 40 cells solved Key:

• Black circles = endpoints of strong links.
• Black lines = strong links.
• Red lines = weak links.
• Green circle = elimination target
Forbidding chain for this elimination:
• b5=6 == i5=6 -- i5=9 == i4=9 -- b4=9 == b4=5
• forbids b5=5
This is a set up for the final forbidding chain.

`     `

### Depth 4 forbidding chain at 40 cells solved Key as before.

Strong sets considered:

1. e28 = 6
2. e85 = 4
3. ea5 = 5
4. ac2 = 5
This type of chain is generally hardest for beginners to find, as it only uses strength in location.

Forbidding Chain Representation:

• e2=6 == e8=6 -- e8=4 == e5=4 -- e5=5 == a5=5 -- a2=5 == c2=5
• forbids c2=6
Now the puzzle solves with Unique Possibilities to the end. (UP 81)

### Solved Puzzle Here is a complete proof in my usual style:

1. Start at 22 filled - the given puzzle. Unique Possibilities to 27 filled. (UP 27).
2. Hidden pair 36 at gh7 forbids ghi9=3,e7=6, gh7=247 UP 30
3. Hidden pair 29 at g12 forbids g46=29, g12=68 UP 37
4. Y wing style: e3=1 == d1=1 -- d1=6 == c1=6 -- c1=4 == b3=4 forbids e3=4 and forbids b3=1 UP 40
1. Locked 4's at def5 forbids e4,ef6=4
2. Y wing style:b5=6 == i5=6 -- i5=9 == i4=9 -- b4=9 == b4=5 forbids b5=5
3. e2=6 == e8=6 -- e8=4 == e5=4 -- e5=5 == a5=5 -- a2=5 == c2=5 forbids c2=6 UP 81
• sets: 2+2+3+1+3+4 = 15
• Max depth 4 at step 5.3
• Rating: .01 + 2(.03) + 2(.07) + .15 = .36 