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