Solving Tips
Sudoku Hidden Triples Technique: Advanced Hidden Pairs Application
Hidden Triples is an advanced version of Hidden Pairs, and is a more complex method among intermediate Sudoku techniques. The core principle is: when three candidates appear only in the same three cells within a unit (row, column, or box), these three cells must contain these three numbers, so you can eliminate all other candidates from these three cells.
Core Principle:
If in a row, column, or box, three candidates (such as 2, 5, 6) appear only in three specific cells, then these three numbers must occupy these three cells. Even if these three cells have many other candidates, all these other candidates must be eliminated, because these three cells can ultimately only contain those three "hidden" numbers.
If in a row, column, or box, three candidates (such as 2, 5, 6) appear only in three specific cells, then these three numbers must occupy these three cells. Even if these three cells have many other candidates, all these other candidates must be eliminated, because these three cells can ultimately only contain those three "hidden" numbers.
Before reading this article, it's recommended to first understand the Sudoku naming conventions for rows, columns, and boxes and the Hidden Pairs technique, which will help you understand the analysis examples below.
Example 1: Hidden Triples in a Row
Let's look at the first example where we find Hidden Triples in Row 3.
Figure 1: In Row 3, candidates 2, 5, 6 appear only in A3, B3, H3, so these three cells must be 2, 5, 6
Analysis Process
1
Observe Number Distribution: Carefully examine Row 3, analyzing the distribution of each candidate. We find that candidates 2, 5, 6 appear only in A3, B3, and H3.
2
Understand the Principle: Because Row 3 must contain the numbers 2, 5, and 6 somewhere, and only A3, B3, H3 have these three candidates in this row, A3, B3, H3 must respectively contain 2, 5, 6 (these three cells will each contain one of these three numbers).
3
Execute Elimination: Since A3, B3, H3 can only contain 2, 5, or 6, all other candidates in these three cells can be eliminated. As shown in the figure, these three cells also have candidates like 1, 4, 7, 8 (marked in yellow), which all need to be eliminated.
Conclusion:
In Row 3, candidates 2, 5, 6 appear only in A3, B3, H3, so the candidates in these three cells are simplified to combinations of {2, 5, 6}, eliminating all other candidates like 1, 4, 7, 8.
In Row 3, candidates 2, 5, 6 appear only in A3, B3, H3, so the candidates in these three cells are simplified to combinations of {2, 5, 6}, eliminating all other candidates like 1, 4, 7, 8.
Example 2: Hidden Triples in a Box
Now let's look at another example where we find Hidden Triples in Box 4.
Figure 2: In Box 4, candidates 3, 6, 7 appear only in A4, B4, B5, so these three cells must be 3, 6, 7
Analysis Process
1
Observe Number Distribution: Examine Box 4 (left-middle 3×3 region), systematically analyzing the position of each candidate. We find that candidates 3, 6, 7 appear only in A4, B4, and B5.
2
Understand the Principle: Because Box 4 must contain the numbers 3, 6, and 7 somewhere, and only A4, B4, B5 have these three candidates in this box, A4, B4, B5 must respectively contain 3, 6, 7.
3
Execute Elimination: Since A4, B4, B5 can only contain 3, 6, or 7, all other candidates in these three cells can be eliminated. As shown in the figure, these three cells also have candidates like 1, 2, 8 (marked in yellow and green), which all need to be eliminated.
Conclusion:
In Box 4, candidates 3, 6, 7 appear only in A4, B4, B5, so the candidates in these three cells are simplified to combinations of {3, 6, 7}, eliminating all other candidates like 1, 2, 8.
In Box 4, candidates 3, 6, 7 appear only in A4, B4, B5, so the candidates in these three cells are simplified to combinations of {3, 6, 7}, eliminating all other candidates like 1, 2, 8.
Hidden Triples vs Hidden Pairs
Let's compare the differences between Hidden Pairs and Hidden Triples:
| Comparison | Hidden Pairs | Hidden Triples |
|---|---|---|
| Numbers Involved | 2 candidates | 3 candidates |
| Cells Involved | 2 cells | 3 cells |
| Identification Feature | Two numbers appear only in the same two cells | Three numbers appear only in the same three cells |
| Elimination Target | Remove other candidates from these two cells | Remove other candidates from these three cells |
| Identification Difficulty | Difficult | Very Difficult |
| Frequency | Occasional | Rare |
Why is it Harder to Identify?
Hidden Triples are harder to find than Hidden Pairs because you need to track three numbers' distribution within a unit, and their relationship is often "masked" by many other candidates. For example, three cells might have candidates {1,2,4,6,7,8}, {1,2,5,6,7,8}, {1,2,4,5,6,8}, which looks chaotic, but careful analysis reveals that numbers 2, 5, 6 appear only in these three cells.
Hidden Triples are harder to find than Hidden Pairs because you need to track three numbers' distribution within a unit, and their relationship is often "masked" by many other candidates. For example, three cells might have candidates {1,2,4,6,7,8}, {1,2,5,6,7,8}, {1,2,4,5,6,8}, which looks chaotic, but careful analysis reveals that numbers 2, 5, 6 appear only in these three cells.
How to Find Hidden Triples?
Finding Hidden Triples requires systematic and patient analysis:
1
Select Target Unit: Choose a row, column, or box to analyze, prioritizing units with more candidates and complex situations.
2
Record Candidate Distribution: For each candidate (1-9) in that unit, carefully record which cells they appear in. You can use paper and pencil to record.
3
Search for Triples: Find three numbers that appear only in exactly the same three cells. Note: These three numbers don't need to appear in every cell, they just need to be limited to these three cells.
4
Confirm and Eliminate: After confirming you've found Hidden Triples, eliminate all other candidates from these three cells, keeping only these three numbers.
Important Notes:
- It must be three numbers appearing only in exactly the same three cells
- If numbers 2, 5 appear in A3, B3, H3, while number 6 appears in A3, B3, C3, H3, they do not form Hidden Triples (number 6 has a wider distribution)
- These three numbers don't need to appear in every cell; for example, A3 might only have {2,5}, B3 might have {5,6}, H3 might have {2,6}
- Hidden Triples are very subtle and require careful, systematic analysis to discover
- It's recommended to use candidate marking features, making it easier to track number distribution
Variations of Hidden Triples
Hidden Triples can appear in different forms:
- Complete Type: Each cell contains some or all of these three numbers. Example: {2,5,6}, {2,5,6}, {2,5,6}
- Distributed Type: The three numbers are distributed across the three cells. Example: {2,5}, {5,6}, {2,6}
- Mixed Type: Some cells contain all three numbers, some only contain part. Example: {2,5,6,8}, {2,5}, {5,6,7}
Regardless of the form, the key is that these three numbers appear only in these three cells, not appearing in other cells of that unit.
Technique Summary
Key points for applying the Hidden Triples technique:
- Observation Dimension: Observe from the perspective of number distribution, tracking the appearance positions of three numbers
- Identification Condition: Three candidates appear only in the same three cells within a unit
- Elimination Target: Remove all other candidates from these three cells
- Analysis Method: Requires systematic, patient tracking of each candidate's distribution within the unit
- Identification Difficulty: Harder to find than Hidden Pairs, requires more detailed observation
- Practical Value: In complex difficult puzzles, may be the key technique to break through bottlenecks
Advanced: Naked Triples Comparison
The counterpart to Hidden Triples is Naked Triples: when three cells in the same unit have candidates that are all subsets of the same three numbers (like {2,5}, {5,6}, {2,6}), you can eliminate these three numbers from other cells in that unit.
Key Difference:
- Naked Triples: Look at cell candidates, eliminate these three numbers from other cells
- Hidden Triples: Look at number distribution, eliminate other candidates from these three cells themselves
Practice Now:
Start a Sudoku game and try using the Hidden Triples technique to simplify complex candidates! Choose a row, column, or box with many candidates, systematically analyze each number's distribution, and see if you can find hidden triples. It's recommended to master Hidden Pairs first before attempting to find Hidden Triples.
Start a Sudoku game and try using the Hidden Triples technique to simplify complex candidates! Choose a row, column, or box with many candidates, systematically analyze each number's distribution, and see if you can find hidden triples. It's recommended to master Hidden Pairs first before attempting to find Hidden Triples.