The following is an illustrated proof for the
Tough Sudoku of February 11, 2007.
Since this is an easy tough puzzle, it is perfectly well suited for study of the limited
techniques required. If quick and easy solving of these types of tough puzzles is your goal,
then this page may be of interest to you.
Since this proof uses a Locked candidate, some
Hidden Pairs, a Hidden Triple, and a Naked Pair,
you may wish to refer to previous blog pages, although it probably is not required. Links to
these pages are found to the right, under Sudoku Techniques.
At many times during this illustration, there are other steps available. It is not the goal
of this page to show every possible step, but rather to illustrate steps that, taken together,
unlock this puzzle
Puzzle at start
A few Unique Possibilities:
- b6 = 3% box & row
- f1 = 4% box
b6 = 3% box & row means b6=3 because it is the only possible place for 3 in both
box b5 and row 6.
f1 = 4% box means that f1=4 because it is the only possible place for 4 in the box.
Singletons in a container (house) are also called Unique Possibilities.
Puzzle at 25 cells solved (UP 25)
Here, there are no remaining Unique Possibilities. It is at this point that I recommend,
highly, that one look for hidden sets before entering all the possibilities. The reason for
this recommendation: Hidden sets are most easily uncovered without all the possibilities
hiding them.
Perhaps the reason they are called hidden is merely because they are obscured by the possibility
matrix.
The possibility matrix is a great tool, but one should be aware that it obscures
deductions one can make based upon location of candidates.
Hidden Pair 13
Illustrated to the left is the existence of a Hidden Pair with candidates 13 at cells
d1 and d3. The only two locations left for both 1 and 3 in box e2 are those cells, d13.
Therefor, if something other that 13 were to exist in one of those two cells, there would
not be enough places left for either 1 or 3 in box e2.
Perhaps fill in cells like this first in the possibility matrix. Remember with a mark, or
a mental note, to not fill any other candidates into those cells. Also, one can use the Locked
candidates nature of 13 at d13 to not fill in any 1's or any 3's at the rest of column d,
(cells d4, d5, d8, d9 - called d4589 for short).
Hidden Triple 349
Illustrated to the left is the justification for placing a Hidden Triple using candidates
349 at cells g8, g9, h9. Hidden triples are rare, but in this case, the alignment of 349 in
column i and the alignment of 349 in row 7 easily reveal this hidden triple. This hidden triple
is the likely reason for this puzzle being called Tough. The logic for a Hidden Triple
is analagous to the logic for a Hidden Pair.
You may note that above I have not filled in each of 349 at each of the hidden triple cells.
The reasoning for this is the 4 already at a8 and the 3 already at h2.
Hidden Pair 49
Illustrated to the left is the justification for a Hidden Pair in row 5. This one is not easy
to find at all. If you were to fail to see it, there will be a naked quad in row 5 available
after entering the possibilities that will accomplish the same eliminations.
The Possibility Matrix filled in with Locked 2's shown
Illustrated to the left is the Possibility matrix after using the information from the
previous steps. Also illustrated are Locked 2's in row i at i89. Because 2's must exist in
box h8 either at i8 or i9, they cannot exist at i5 or i1.
This isolates the 2 at a5 as the only possible 2 in row 5. Therefor,
Puzzle at UP 26
After cell a5=2, both cells
b3 and b4 are limited to only 47. This is a naked pair. Since these two cells are limited to 47,
one can safely eliminate 47 from the rest of column b. Thus:
- Pair 47 at b34 forbids
- forcing
- c8 = 7% box
- f6 = 7% row
- % cell to the end
- UP 81
Solved Puzzle
Proof
Below is a proof written in my usual style. Only required steps are listed.
- Start at 23 filled - the given puzzle. Unique Possibilities to 25 filled. (UP 25).
- Hidden pair 49 at bh5 forbids b5=2, h5=126
- Hidden triple 349 at g89h9 forbids g8=12, g9=126, h8=1268
- Locked 2's at i89 forbids i15=2 UP 26
- Pair 47 at b34 forbids b5=4,b17=7 UP 81
- Sets: 2+3+1+2 = 8
- Max depth 3 at step 2.2
- rating: .14 - not too tough
Summary
Hidden Sets are a valuable search to perform before entering the possibilities.