Similar to Block Elimination Technique, Combination Elimination Technique is an advanced technique in Direct Elimination Techniques, but it is rarely used in practice. Therefore, it is difficult even to find a proper example to illustrate this technique. However, if you wish to use this technique in a high priority, you can still find a lot of cases which meet the conditions it requires.
From its name, we can learn that this technique considers some kinds of combinations as the base of eliminations, while these combinations include not only those from blocks, but also those from cells. For example,
We want to determine the exact location of digit 6 in Block at [G4]. It seems quite difficult at first glance. Although two 6s at [G9] and [H3] can help us exclude [G4] and [H6] in Block at [G4], we are still unable to determine whether 6 should be assigned to [I4] or [I5]. So now it's show time for Combination Elimination Technique.
First let's put the Block at [G4] aside and take a look at the two blocks above it, which are Blocks at [A4] and [D4] respectively. These three blocks have a common point that all of them occupy the same columns, namely Column 4 to Column 6. So the numbers placed in these blocks always affect each other directly.
For Block at [A4], with the help of value 6 at [A1], [A6] can be excluded so that the possible positions of value 6 finally become two cells only: [B5] and [C6].
For Block at [D4], the value 6 at [E7] can exclude [E4] and [E6] and leave only [F5] and [F6] to be the only possible positions for 6.
By now, we are still unable to determine the exact positions for 6 in these two blocks. But let's make some analysis first:
Therefore, there are only two possibilities of combinations: [B5] = 6 and [F6] = 6; or [C6] = 6 and [F5] = 6. No other circumstances. However, whatever combination will occur, there should be an existing 6 in Column 5 and Column 6 respectively and the cells being assigned as 6 must lie in Blocks at [A4] and [D4] respectively. So that in Columns 5 and 6, the rest of the cells are impossible for holding value 6. With this knowledge in mind, you can go on with the puzzle now.
As mentioned earlier, we can use value 6 at [G9] to exclude [G4]. And then we use the combinations above as an input to exclude [H6] and [I5] in Block at [G4], leaving a sole position [I4] as the only possible location for value 6.
Now let's review the example above and see what conditions are to be satisfied if we want to apply Combination Elimination Technique.
Here is another example:
To locate the position of value 1 in Block at [D4], we need the help from its "side-by-side partner blocks".
Value 1 at [I2] can exclude [E2] and [F2] in Block at [D1], leaving [D1], [D3] and [E1] as the possible holders for value 1 .
Value 1 at [H7] can exclude [E7] and [F7] and value 1 at [A9] can exclude [E9] and [F9]. Therefore, in Block at [D7], 1 can only be placed into [D8] or [E8].
Although there are three cells that are possible to be placed with value 1 in Block at [D1], they occupy only Rows D and E, which is exactly the same as in Block at [D7]. So there are only three possible circumstances for the position of value 1 in these two blocks: [D1] = 1 and [E8] = 1; or [D3] = 1 and [E8] = 1; or [E1] = 1 and [D8] = 1. Whichever situation happens, value 1 must occur in Rows D and E. So those cells in Block at [D4], which lie in Rows D and E, have no chances to be assigned as value 1.
Therefore, we can exclude cells [D4], [D6] and [E4] by Combination Elimination Technique. And with the help of value 1 at [G4], we can finally determine that 1 must be placed in [F6].
Read more examples to gain a better understanding of this technique:
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