This technique is used occasionally in practice. It is not because this technique is difficult to understand. Actually you can quickly master it. However, it is really hard to identify the situation to which Rectangle Elimination Technique can be applied. But compared to Combination Elimination Technique, you have even more chances to use it. To illustrate this technique, let's start with an example:
For this puzzle, you cannot go on any more with other Direct Elimination Techniques. We are now trying to find where digit 8 should be placed in Block at [G1]. It seems really difficult to do it. We can use the value 8 at [B2] and [E3] to exclude [G2], [H2], [G3] and [I3] in Block at [G1], but we are still uncertain whether 8 should be assigned to [G1] or [I1].
Let's put aside the Block at [G1] and focus on Column 6 first. There is currently no 8 placed in this column, so where should 8 be placed? We find that the value 8 at cell [B2] excludes [B6], and value 8s at cells [E3] and [F4] excludes [E6] and [F6] respectively. So in Column 6, only [C6] and [I6] are possible to hold value 8. Please refer to the figure below:
Similarly to Column 9, because of the exclusion of 8 at [F4] to [F9], the only possible cells to hold 8 become [C9] and [I9].
Fortunately, in Column 6 and Column 9, the possible positions for value 8 lie in the same two rows, which are Rows C and I. This is the prerequisite for us to use Rectangle Elimination Technique.
If [C6] = 8 in Column 6, then [I6] and [C9] must never be assigned as 8. So that only [I9] in Column 9 is left for holding value 8; Or
If [I6] = 8 in Column 6, then [C6] and [I9] must never be assigned as 8. So that only [C9] in Column 9 is left for holding value 8.
There is no more circumstance besides these. Therefore, either [C6]= 8 and [I9]= 8; or [I6]= 8 and [C9]= 8. No matter which case is the final solution, 8s are placed in both Row C and Row I. All cells in these two rows except those lie in Column 6 and Column 9 cannot be assigned as 8 any more. And this can help us to find a solution.
Now let's look at the Block at [G1]. We have known that for value 8 only [G1] and [I1] are available for our determination. Therefore, based on the analysis above, we can use Rectangle Elimination Technique to exclude [I1] in Row I, so 8 must be assigned to [G1].
Here are the conditions that have to be satisfied before you can use Rectangle Elimination Technique:
Here is another example:
At this stage of the puzzle, Rectangle Elimination Technique is perhaps your only choice to move on. Let's focus on finding where number 4 can be put in Block at [G1]. The figure above shows basic eliminations, which are quite understandable, but how can we determine whether 4 should assign to [G2] or [G3]?
First take a look at Row B. In this row, [B4] and [B5] are excluded by value 4 at [C5] because they are in the same block. In addition, value 4 at [H8] excludes [B8], so [B1] and [B3] are the only possible positions left for value 4.
In Row F, because of the exclusion of 4 at [D6], [F6] is excluded. Therefore we can only place value 4 in [F1] or [F3].
We are happy again to find that all possible cells for value 4 in Rows B and F happen to be in the same columns, which are Columns 1 and 3. All cells in these two columns other than those in Rows B and F cannot be assigned as 4. Therefore, [G3] is excluded as well. So [G2] = 4.
Here are some more examples to help you master this technique:
Rectangle Elimination Technique is the most difficult technique in Direct Elimination Techniques. This is because it is always very hard to find when and where this technique can be applied. So try this technique only when all other techniques fail. And even if Rectangle Elimination Technique cannot help you any more, you can probably try Candidates Elimination Techniques.
See Also:
