Block Elimination Technique is an advanced technique in Direct Elimination Technique. Although it is less used than Basic Elimination Technique, Block Elimination Technique is a good supplement to it.
Block Elimination Technique is based on the relationship between blocks and rows or columns, which is quite similar as Basic Elimination Technique. However, the difference between these two techniques is that Block Elimination Technique eliminates cells not by existing assigned values. Sometimes uncertain positions for a value can also be used to determine the solution to a cell in a unit. To understand this, let's first see an example:
For the above puzzle, we fail to find any more solution to any cell by using Sole Position Technique and Basic Elimination Technique only. Now we try Block Elimination Technique.
We first start from Block [G4] which has least unassigned cells. In this block, only [H6] and [I5] are empty while digits 1 and 2 are missing. We have to determine the exact position for each of the two values.
Since [D2] = 2, by using Basic Elimination Technique, we exclude [H2] and [I2] from holding value 2 since they are in the same column as [D2]. And this makes [I1] and [I3] become the only possible cells for value 2 in Block at [G1]. See the figure below:
Although we still cannot confirm the exact position of value 2 in Block at [G1], we are lucky to find that both cells that are possible holders of value 2 lie in Row I. That is, no matter which cell will actually be placed with value 2, the rest of cells in Row I can never hold 2 any more. So cell [I5], lying in Row I, is excluded. Therefore, for Block at [G4], only [H6] is left for value 2. So we set [H6] = 2. Next, set [I5] = 1 based on Sole Position Technique.
From the example above, we can understand that unlike Basic Elimination Technique, this elimination technique cannot be used directly on the target unit. Helps from other units are required somehow.
There are totally four circumstances in practices:
Case 1and2are most commonly found. The previous example illustrates Case 1.Now let's see an example of Case 2:
In Block at [A7], there are totally four empty cells, but we can still find the position for value 5 easily. Why? Since we find that in Block at [G7], all possible positions for value 5 happen to be in the same column, which is Column 8. The exact position of value 5 in this block is not important now, because the current information is enough for us to exclude [A8] and [B8] from holding value 5. Therefore, in Block at [A7], the possible positions for value 5 are reduced to [A9] and [C9]. By using Basic Elimination Technique, we can exclude [A9] with the help of value 5 at [A4] and finally determine that [C9] = 5.
Now let's go on with some unusual cases:
In Row C, the position of value 3 can be determined as follows:
First, focus on Row B only. By using Basic Elimination Technique, we can exclude [B2] and [B3] through the 3s at cells [H2] and [F3]. That will make [B4] and [B5] become the only possible positions for value 3 in Row B.
Fortunately, both cells happen to be in Block at [A4]. Now the condition listed in Case 3 is satisfied. Other cells in Block at [A4], including [C4] and [C5], are unable to hold value 3 any more. Together with the help of value 3 at [F3], the position of value 3 in Row C can only be [C1].
The example for Case 4 is shown as below:
In this case, we will try to determine the position of value 8 in Column 6 with the help of Block Elimination Technique.
Let's first look at Column 4. The 8s at cells [C3] and [I8] help exclude [C4] and [I4] respectively, leaving [D4] and [F4] the only possible positions in this column for value 8. Luckily, both cells are in the same block and this helps exclude [D6], [E6] and [F6] (If any of these three cells holds value 8, then nowhere in Column 4 can be assigned as value 8). So [B2] becomes the sole position for value 8 in Column 6.
In practice, we can find much more complicated cases. For example,
can you determine the position of digit 3 in the Block at [A1]? Just think about it.
Here is the answer. The value 3 at [C5] not only excludes the possibilities of [C1] and [C3] for holding 3, but also excludes [C8] and [C9] as well. This makes [B8] and [B9] become the only possible positions for value 3 in Block at [A7]. See the figure below:
It is clear that value 3 in Block [A7] must lie in Row B. Therefore, [B1], [B2] and [B3] can all be excluded by using Block Elimination Technique ( Case 1). Now only [A1] and [A2] are left as the possible positions for value 3 in Block at [A1]. But we still cannot determine the exact position for value 3. So we have to search for more help from other units.
Observe the Block at [D1]. The 3s at cells [G3] and [D7] help exclude [D1], [E3] and [F3] by using Basic Elimination Technique. Hence, only [E2] and [F2] are left as possible holders of value 3 in Block at [D1]. These two cells happen to lie in the same block. Therefore, we use Block Elimination Technique ( Case 2) to exclude [A2] as well, leaving [A1] the sole position for value 3.
In this example, multiple assistant blocks participate in the elimination process. Although this kind of situation is not common in practice, it can be seen sometimes. The key to utilize this technique lies in correctly identifying the condition and properly applying different elimination techniques.
Below are more examples to help you understand and master this technique:
See Also:
