Mastering Sudoku: Discovering Hidden Triples in 9×9 Puzzles

Have you ever wondered how some people are able to solve Sudoku puzzles with such ease?

One of the key strategies in mastering this popular number puzzle game is to understand and utilize hidden triples.

We will explore what hidden triples are, how to identify them, why they are crucial in solving Sudoku, and provide you with step-by-step guidance on finding hidden triples.

Stay tuned for examples, tips, and common mistakes to avoid in your Sudoku solving journey.

What is Sudoku?

Sudoku is a logic-based combinatorial number-placement puzzle that has been referred to as a Latin square according to its simple algebraic properties. It became known as Sudoku in Japan in 1986 and helped tune analytical skills, improve patience and enhance memory. It then became a worldwide popular puzzle allocated as a pastime to kill time while waiting or since it was available in daily newspapers from the early 1970s, to be solved during leisure time.

Mathematically speaking, a Sudoku is a discrete function from a finite set called the grid onto another finite set called glyphs. A Sudoku puzzle consists of a 9×9 grid that is partitioned into 9 3×3 grids. The goal is to fill in the grid so that every 3×3 grid, row, and column contains the digits 1 through 9.

A Sudoku puzzle grid consists of 81 cells evenly divided into 3×3 subregions, 3 columns, and 3 rows, with each of them containing 9 cells. Each Sudoku puzzle starts with some cells pre-filled with a number or a glyph from the finite set of valid entries, as shown in the following figure. The player then solves the Sudoku puzzle by using the basic strategies of crosshatching, number elimination, finding hidden pairs, triplets, and quads, and the advanced strategy of Alternating inference chains (AIC).

How to Play Sudoku?

To play Sudoku, fill the grid where the numbers 1-9 appear only once in each row, column, and 3×3 box. Whether you play using hidden triples or another technique to solve the puzzle is up to you.

To complete a Sudoku grid, observe the patterns from all 3 perspectives. Always find all possible locations first. Look for the interplay between blocks and rows. Explore cleared cells before transitioning to another area. If one approach stalls out, switch back to another pattern. Do this by looking at the way blocked cells impact the cleared ones.

Pay attention to how pre-filled values in unused blocks clue us into row or column values. Hidden singles or pairs might soon reveal themselves. Working section by section increases the usefulness of the hidden triple method. This way players are not just focusing on finding the triple itself, but can now leverage the hidden triples they do find.

What are Hidden Triples in Sudoku?

Hidden triples in sudoku refer to a situation where the members of a triple are interacting together, but two or more cells are candidates for one or more of the triple candidate digits. If those digits appear in the two or more cells, then the triple would not be a hidden triple. They hide their identity and appear to be in other interactions such as hidden pairs or naked pairs.

Hidden triples are in this way more difficult to find than naked triples, even though they may have only the minimum three pairs. Two ways that triples are able to hide themselves are by using hidden singles (a pair of two cells which are interacting) and pairs (which involve a situation of at least 4 cells). In the following example, triples {2, 3, 9} could now appear in S5C3, but the digit sequence is not yet short enough to prove a hidden triple exists between {2, 3, 9} and {4, 6, 8}.

What are the Rules of Hidden Triples?

The rules of Hidden Triples in Sudoku are that there must be exactly three candidate locations in three different rows, columns, or boxes. Each candidate location must be able to see all the others. Those numbers must not appear elsewhere in the same row or column, and they must be the only numbers in the naked subset to have the same three rows, columns, or boxes. Those cells must also necessarily be part of a legal solution to that particular puzzle. These triples may contain additional candidate numbers, but there must be only three unique candidates.

How to Identify Hidden Triples?

Hidden triples are identified by starting to spot cells that cannot have any more candidates, then using that information to eliminate candidates from the same row, column, or block, so that a pattern is revealed. When first given a puzzle, the right course of action is to work on the naked and hidden singles, combinations, and other patterns so that the possibilities for cells with fewer than three candidates are determined.

The key to learning hidden triples, quadruples, and other advanced solving methods is the systematic elimination of candidate numbers. If no additional information is provided by other solving methods, Sudoku tripling is likely the next step. This strategy can eliminate candidates for some cells through the careful observation of other cells. This is particularly so for the row or column in which a triplet is discovered as a result of a hidden pair in a block.

Using the same example as earlier for a Hidden triple in cell A4 in this column specifying the cells), some Sherlock aspects of this particular strategy include the following:

• 1 in A2 eliminates 1 in A9 and 1 in G4
• 1 in A9 eliminates 1 in A2 and 1 in G4

Why are Hidden Triples Important in Solving Sudoku?

Hidden triples are important to solve Sudoku because they narrow down the state space during the logical deduction process for solving the puzzle. State space in this scenario refers to the number of containable cell possibilities in the grid, given a certain combination of numbers. The broader the state space, the more likely it will be that incorrect initial assumptions will be made. Conversely, the smaller the state space, the more certain it is that the assumptions are correct and the current cell possibilities are easier to prove correct or incorrect via constraint elimination.

Hidden triples are technically not needed to solve a Sudoku puzzle, but they are a good way to straddle the middle of the road between having too broad a decision space versus too limited of one. Having a good balance in this regard will help you make fewer errors to get to the triggered assumptions and use them to quickly eliminate other numbers. This naturally leads to faster puzzle solving, as the hidden triples technique helps you mentally configure the board closer to a solvable state.

How to Find Hidden Triples in Sudoku?

To find hidden triples in sudoku puzzles, you have to look for three candidate numbers residing in only three cells of a row, column, or box. If there are no additional candidate numbers in these cells, these three are hidden triples.

The first step in identifying hidden triples is to eliminate all the possibilities of there being naked or indicated triples (if there are more than three open cells in a row, column, or box that share three identical numbers). If those cases do not exist, you have found a hidden triple.

Let’s find a hidden triple in Wikipedia’s example puzzle. In row C, you can see that the only cells with the candidate numbers 3, 4, 8 are the blue-pathed squares:

• C1: 4
• C7: 8
• C8: 3

Cells C2, C4, and C6 all have numbers, so they are not part of the triple. Row C has a hidden triple of 3-4-8. The benefit of hidden triples is that, though considered rarer than other types of triples, it may assist in eliminating possibilities for easier cells to mark.

With row C, cells C7 and C8 are correct, which implies that cell C1 must complete the triple. This makes it easier to eliminate the potential false positive of hidden pairs you’ve created when marking candidate numbers in early stages.

Step 1: Identify Potential Candidates

To begin finding hidden triples in a sudoku puzzle, the first step is to identify candidate cells with the unique properties necessary for linked cells. When considering the 27 combinations of 3 unique 3 by 3 boxes in the grid, there are 840 naked triple combinations. Identifying each of them early on can cut down on potential branches to explore before needing to backtrack.

To determine candidate cells or hidden triples, a sudoku player should first go through all of the cells in the puzzle to determine the potential naked triples and the candidate cells that could form those triples. You can then cross out cells which cannot be candidates (since they are already used in the row, column, or box). During this part, it is fine to include cells that are part of a pending hidden triple, even if the triple itself contains more than 3 cells.

In the following 4 by 4 grid, the cells that can be marked as linked for hidden triples are marked with a red square. A player would go through and identify as many of the following as possible, but as of this step of the strategy it is fine to include linked squares from different groupings.

Step 2: Look for Overlapping Candidates

Look at the intersection of all combinations of three Strong Links formed. This will indicate which squares can become the three members of the Hidden Triple. If the three candidates are in the same row, those three numbers do not appear in any other candidate squares within the same row. The same logic is applied to a column or a box.

Step 3: Eliminate Non-Candidates

1. Highlight hidden triple candidates in each box of the respective 3-row, 3-column subregions (quotes 3): Boxes 2, 3, 4, 5, and 6 have only 3 remaining candidates at cells that line up in a row or column, the value of the 3 candidates in those boxes is hidden triple candidates and should be highlighted.

Step 3 of finding hidden triples in Sudoku is to eliminate non-candidates. Hidden triple candidates in cells of each identified three-mini-row or column that are not those three values are eliminated from consideration. Additionally, any hidden triple candidates in identified three-row or columns (macros) that are not three different values are eliminated from consideration. If the processes in steps 1 and 2 have been carried out correctly, the identification of hidden triples should by now really be a hidden set of triple values.

Step 4: Identify the Hidden Triple

If there are even 2 candidates that appear in the 3 cells, a hidden triple exists. When they are recognized, an eliminator is applied, removing their possibilities from other candidate cell blocks. It is also often a corollary that if a hidden triple is identified, at least one of the numbers in the triple must be in one other cell row, block, or column. Apply the famed Triple-Double Interchange if you are having trouble.

Examples of Finding Hidden Triples in Sudoku

Here are four images showing the techniques of finding hidden triples in Sudoku.

• A simple 4-empty hidden triple case in a 9×9 grid.
• An intersection chain hidden triple example of 2 clusters of 3 cells in an 8×8 grid.
• A near-naked triple case of 3 hidden triples in a 9×9 grid.
• A monster Sudoku grid with 8 hidden triples. Note this image mostly shows the initial grid with some EPD operations. At the start, Hidden Triple-C is already easily identifiable.

Tips for Finding Hidden Triples in Sudoku

Three tips for finding Hidden triples in Sudoku are:

1. Merge the Hidden Triple visual exercises with row, column, and block exercises.
2. Only look for Hidden Triples after solving easy pairs and triples.
3. Avoid Hidden Triples if they lead to too many possibilities.

Tip number 1 refers to exercises specifically combining several principles and methods of Sudoku. Number 2 refers to hidden triples merely being one category of hidden pairs and hidden singles. And number 3 means that more complex deductive reasoning, if not accompanied by step-by-step progress, should not be used since an incorrect choice during the exercise may derail the solution. To play with Hidden Triples and get into the right mindset, use the Hidden Triple exercise present on the Sudoklue website.

Common Mistakes to Avoid when Finding Hidden Triples in Sudoku

The most common mistake to avoid when looking for hidden triples in sudoku puzzles is looking for pairs. Hidden triples are found when examining 3 possible triple combinations of candidates, but mistaking a pair makes the correct triple difficult to find. The more complex the puzzle is, the more increased accuracy is required.

Additionally, do not use computer systems to assist you when you’re learning to utilize hidden triples. This can come later when you are more proficient. On one hand you could miss the opportunity to improve your logical reasoning by letting AI solve the most difficult puzzles. An even worse consequence is the automation chooses wrong combinations which then might be difficult to find. Regularly training the usage of hidden triples helps strengthen cognitive muscles in a fun and interesting way! If the assistant says the there is no hidden triple in sight, take a quick screenshot of the board and start advanced strategies. It’s amazing how frequently there is a hidden triple available that is being missed.

Conclusion

Stage 1 and 2 of most sudoku puzzle-solving methodologies provide the most straight-forward of practical methodologies to find hidden triples in sudoku. Hidden triples often give the most progress in completing the puzzle in a minimal amount of time. They are not the last numbers in a row, column, or sector, but finalize the selection of the triple by removing the candidates. Practical application is the most important way to understand and recognize hidden triples.

Frequently Asked Questions

What are hidden triples in Sudoku and why are they important to find?

Hidden triples are three numbers that appear in the same row, column, or block in a Sudoku puzzle, but are not immediately obvious. These triples can help to eliminate other possibilities and solve the puzzle faster.

How do I identify hidden triples in a Sudoku puzzle?

Look for three numbers that only appear in the same row, column, or block and narrow down the possibilities based on that information. If those numbers are the only possible options for a certain square, it is a hidden triple.

Are hidden triples always easy to spot in Sudoku?

Not always. Sometimes hidden triples can be more difficult to identify, especially if there are multiple possibilities in a row, column, or block. It requires careful observation and elimination to find them.

Can hidden triples be found in any Sudoku puzzle?

Yes, hidden triples can appear in any Sudoku puzzle. They are a common solving technique and can be found in easy, medium, and difficult puzzles.

What strategies can I use to find hidden triples in Sudoku?

One method is to scan each row, column, and block for three numbers that only appear in those spaces. Another strategy is to use elimination and deducing to narrow down the possibilities and find hidden triples.

How can hidden triples help me solve a Sudoku puzzle faster?

By identifying hidden triples, you can narrow down the possibilities for certain squares and eliminate other options, making it easier to fill in the remaining numbers and complete the puzzle. This can save time and make the solving process more efficient.