The other day, while driving I came upon a traffic signal. The red light was clearly lit, but
so was the green. I was confused. And a little bit scared. A few more brown strands turned gray.
Suppose you are working hard on a puzzle. You get to the end. You see that no matter what,
you cannot uniquely identify the solution. The puzzle has two (or more) valid solutions!
Puzzle solvers are enamored by the latter situation to the same extent drivers are endeared
to the former.
For this reason, my opening blog codified this rule, practiced by all good puzzle creators:
- Each Sudoku puzzle has a unique solution
This
rule has some power attached to it. It is necessary, however that one has digested
the previous techniques to apply this idea.
The simplest application of the unique solution rule is Unique Rectangles. However,
there are some extensions of this idea that, to date, I have not found very useful. If a puzzle has
no more than one solution, then
- All possible partitions of the puzzle grid that have multiple solutions are forbidden
That seems fairly obvious. The specific application of this idea with
Unique Rectangles
follows:
- The following configuration is forbidden
- A single naked pair of candidates x,y exists in a box and a row
- Another naked pair of candidates x,y exists in a different box and row
- The column coordinates of both of these pairs are the same.
- Therefor, any potential solving activity that would cause this forbidden configuration to occur
is also forbidden.
Proof:
- Suppose the above configuration occurs
- Then because of the naked pairs in two rows, two columns, two boxes
- The puzzle is now partitioned into a large puzzle and a small subpuzzle
- The small subpuzzle is never effected by the larger puzzle
- The small subpuzzle has two solutions
Here is an example:
Note here that we have pair 17 in rows 1,3 and columns a,i and boxes b2,h2. Therefor, nothing
can happen to prefer a1=1 over a1=7. Then same is true for cells a3,i1,i3. Thus, the puzzle
has at least two solutions that do not violate the standard rules.
The power in this technique, then, is that when one has an Almost Unique Rectangle,
one can forbid something. I prefer the term forbidden rectangle over unique rectangle,
but the nomenclature is fairly standard with this technique, so I guess it best to call this situation
Unique Rectangle.
Here is an example of the simplest usage of this concept
In this example, clearly if a1<>9 then we would have a Unique Rectangle. Therefor,
a1=9. This step could be written in a proof as:
- AUR 17 at ai13 forbids a1=17
False Forbidden oops! Unique Rectangle
The example above reflects a common error. The situation above is not a Forbidden Rectangle.
Note that although it looks very much like one, none of the pairs 23 share a common box. Thus, something
could happen, say at cell c2, that would break the rectangle legally and not force two solutions.
Twelve Solutions
There are many extensions of this idea to other shapes and configurations. I can say that
except when trying to create a puzzle, I have never encountered any of them except BUG.
The Bivalue Universal Grave may be treated in a later blog. A configuration like
that shown above would also be forbidden, as it would have at least two solutions.
Deeply hidden Unique Rectangle
Can you find the Almost Unique Rectangle here? It is very well hidden.
If only life were as unambiguous as Sudoku!