Keith's conjecture: For every Type 3 reduction there is a Type 4 reduction that achieves the same result.
Consider the following:
- Code: Select all
+--------------+
| - - - |
| ab - ab |
| - - - |
+--------------+
| abc - abd |
| - - - |
| - - - |
+--------------+
Others have named <abc> and <abd> the "roof cells".
The non-unique "deadly pattern" is:
- Code: Select all
+--------------+
| - - - |
| ab - ab |
| - - - |
+--------------+
| ab - ab |
| - - - |
| - - - |
+--------------+
and <a> and <b> are the "deadly values".
One unique solution fragment would be:
- Code: Select all
+--------------+
| - - - |
| ab - ab |
| - - - |
+--------------+
| c - d |
| - - - |
| - - - |
+--------------+
which is fine. Except, if this is indeed the case, neither a Type 3 nor a Type 4 reduction would be possible.
Both the Type 3 and the Type 4 reductions are possible only if, in the solution, one of the roof cells is a deadly value.
The Type 3 reduction is this:
- Code: Select all
+--------------+
| - - - |
| ab - ab |
| - - - |
+--------------+
| abc - abd |
| - - - |
| - cd - |
+--------------+
One of the roof cells must be <c> or <d>. Together with <cd> the roof cells form a pair, so <cd> can be eliminated from all other cells in the house. But then, one of the roof cells must be <a> or <b>.
The Type 4 elimination says that the roof cells are a strong link on one of the deadly values, so the other can be eliminated from the roof cells.
As a practical matter, it seems that if a Type 3 reduction is possible, an easier to recognize Type 4 is also there.
I am not proposing the reverse: That every Type 4 has a companion Type 3..
Keith
