Intermediate Sudoku Strategies: Triples Mystery Cache

The description may seem very long. However, I have tried to get very granular. If you would like to skip right to the puzzle, please click here.

There are two main intermediate strategies for solving Sudoku, Triples and Ghost Numbers. By using those two strategies alone (along with the basic ones), you can solve any puzzle in the newspaper except for possibly a few of the Saturday ones. Well, you can solve the ones that are solvable. A few of the Saturday ones have multiple solutions and really can’t be solved by “normal” means. Here I address Triples.

Let’s start very, very basic. Note this diagram. You can see that the yellow cells must contain 1-2-3. We don’t know the order, but we do know that no other digits can be put there. Now for the fundamental deduction: Since the yellow cells can only contain 1-2-3, none of the orange digits can contain 1-2-3 either.

That’s all there is to it—in theory. However, triples can be very sneaky little devils and hide in the oddest places. The advanced newspaper Sudokus (say, Thur-Fri) all take the sneakiness to new levels. The issue is that while Doubles (which I classify as a basic strategy) involve only one combination, Triples involve eight! Throw in three dimensions (row, column, and box) and those eight combos can be lurking where you least expect them.

Before we get too far, we need to work up to the eight combos. The diagram above shows only one, the easiest one to spot. Let’s add a few digits to one of the columns, taking the 3 out of the picture in one of the yellow cells. Note the diagram below. The yellow cells still contain only 1-2-3 and 1-2-3 can only be put into the yellow cells, even though one of the cells can only contain 1-2. Thus, not only can the Triple be three cells with 1-2-3 in each, but also they could contain (1-2-3, 1-2-3. 1-2), (1-2-3, 1-2-3, 1-3), or (1-2-3, 1-2-3, 2-3). That makes four combos to those of you keeping score.

Let’s add a few more digits to another column. Now we still can only have 1-2-3 in the yellow cells, preventing them from being in any of the orange cells, even though one cell contains 1-3 and the other 1-2. Three more combos: (1-2-3, 1-2, 1-3) (1-2-3, 1-2, 2-3), and (1-2-3, 1-3, 2-3).

Now for the rabbit out of the hat! Let’s add more digits to the third column. We still have three yellow cells that can only contain 1-2-3, even though none of the individual cells contains all three digits. That is our eighth combo: (1-2, 1-3, 2-3). Someone with a combinatorics bent may now want to ask, “What about (1-2-3, 1-2, 1-2)?” or something similar. Think about it. With that combination, you have a double with the two (1-2)’s and the third cell must contain a 3.

Now let’s use this information to progress a little farther in this puzzle. What’s next? Well, I can’t tell you what’s next. The reason is that there are at least three different things you can do next. They all lead to the same result, but different people do them in different orders. It’s all a matter of personal taste, what techniques you know, what techniques you like, what you notice first, or where you start in the puzzle.

Here’s one way to proceed. Notice the second column. Once you’ve discarded the 1-2-3 from the top middle square, you will notice that you have another triple (!) in the blue cells. That means you can discard the 6-9 from the green cell. At this point you may notice that the two underlined cells are the only place in their column where a 6 can appear. Therefore they CANNOT appear in the tan cells and you can discard the 6’s from them. You now have yet another (!) triple in the right column, allowing you to get rid of the 9 from the 1-2-6-9 cell in that column, leaving another triple in that column (1-2, 1-2-6, 1-2-6).

Got that? Let’s back up. Here is the same grid after dealing with the 1-2-3 triple.

Some people will look at the first column instead of the second and notice that the 23-239-239 form another (!) triple. Their next step is to remove the 2’s, 3’s, and 9’s from the rest of that column. This means that the green cell is a 7. Furthermore, since the blue cells are the only place in their column where a 9 can be, they can remove 9’s from the rest of that 3 x 3 square. We now have yet another (!) triple on 1-2-6 in the three underlined cells and can remove the 6’s from the two 5-6-7-9 cells at the top of the column.

Notice that this sneaks around to the conclusions formed in the previous analysis. Notice too that putting the 7 in the green cell allows you to remove the 7 from the 5-7-9 tan cell. And NOW (!!) you have a 5-9 double in that row. This allows us to remove all 5’s and 9’s from the squares with the little xx’s in the same row. This sort of analysis usually allows you to solve the rest of the puzzle.

Now, let’s back up a THIRD time, but this time we’re going back even farther than where I started at the beginning of this puzzle. Here is the grid before I started filling in colored squares and itty-bitty pencil marks.

Some people won’t start with looking for a triple at all. These people will look at this pattern and notice that there is a 4 and an 8 in the third row and also in the 5th and 6th columns. Note the red arrows. This means that the two cells with the blue circles can ONLY contain a 4 and an 8. (In Sudoku lingo this is often called a Hidden Double.)

If you are this sort of person, your next step will probably be to notice that the two 7’s prevent a 7 from appearing in the cells with blue X’s. This means that the 7 in that column has to be in the blue cell. Then you’d probably notice that the two tan cells can only contain a 5 and a 9. Then you’d probably start eliminating 5’s and 9’s from that row and continue from there. You haven’t touched a triple at all. Yet. You’d probably hit a wall at some point and then start looking for triples or anything else that strikes you.

The point here is that there is no single way to approach a Sudoku. This is part of their intrigue. You start doing the “obvious” (obvious to you) things and then start pulling out various other techniques when you run out of the “obvious” stuff. This is also why it is so difficult for me to construct these puzzles. I can work up an elaborate Sudoku thinking that you MUST eventually get to the teaching example—only to find that there is another way to work the puzzle and you can bypass the example altogether.

Here are the ingredients for spotting a Triple.

1. A triple appears in exactly one row, column, or 3 x 3 square.

2. There must be exactly THREE cells in that row, column or square such that...

3. Exaclty THREE digits and only those digits can appear in those cells.

These three digits are locked to these three cells. You can then eliminate ANY OTHER occurrences of these three digits from the OTHER cells in the SAME row, column, or square.

(Quick Tip: If the THREE cells exist in both a row/column and a square you have a bonus! You can eliminate the digits from both the row/column and the square.)

[Email me if this doesn't make sense.]

Before I continue on to the puzzle and the cache, I want to point out the logical extension to triples: quads and quints. The same analysis shown above can be applied to a set of 4 (or 5 cells) in the same square, row, or column. Quads are not too common, but they do exist. However, often you can usually find something else to avoid having to resort to looking for them. After a little bit of working with triples, you’ll pick up the technique and be able to handle quads with ease. A true quint on the other hand is very rare. Usually, you will be able to find a double, triple, or quad in the other 4 cells and be able to ignore the quint altogether.

Below is an example of a quad adapted from a computer program I have. The blue cells contain only the numbers 1-5-6-9 even though only one cell has all four digits and the other three only have two digits. However, the fact that all four cells are in the same 3 x 3 square allows you to eliminate those digits from the other four open cells in that square.

Here is another example adapted from one by Tetsuya Nishio. This highlights the different ways that people can approach a puzzle. Look at the first column. Some people will note the quad on 4-6-7-8. This allows you to remove those digits from the other three cells. (This is called a Naked Quad.) Others will use a slightly different analysis. They will note that 2-3-9 exist in the top left 3 x 3 square. They will also note that there is another 2-3-9 in the last row. If you combine these two occurrences, there are only three cells remaining in the first column which can possibly contain the 2-3-9, meaning that they MUST contain the 2-3-9. (This is called a Hidden Triple.) Either analysis yields the same result. Furthermore, once you’ve made these deductions, you now know that the red circle must contain a 6.

Now for the puzzle and cache!! (Remember? That’s why you started reading this diatribe in the first place.) Here are three grids adapted from a computer program I have. Each has (or rather, should have) exactly one useful triple. A useful triple is one that allows you to remove digits from other cells. (The math geek crowd may prefer the term “nontrivial triple.”) Example. In the first grid, the three empty cells in the sixth row from the top (similarly in the eighth row from the top) form a triple, but the example is not useful. Noticing it doesn’t allow you to progress in the puzzle. You have to find a useful triple.

Once you find the triples, add the digits together. Take the final digit if the result is a two-digit number. That is the digit to plug into the equations below. Example: In the very first diagram above, the yellow cells form a triple on 1-2-3. The sum of their digits is 6. In the fourth diagram, the blue cells form a triple on 5-6-9. The sum of their digits is 20 so you would work with 0. The first grid gives you X; the second, Y; and the third Z. The cache is at N 37 33.ABC, W 122 18.DEF where

A = X + Y

B = Y

C = X

D = Z

E = Y + Z

F = Z

Here is the puzzle to derive the X digit.

Here is the puzzle to derive the Y digit.

Here is the puzzle to derive the Z digit.

In each case, please remember to take the final digit of the sum.

One note of caution. As I pointed out above, you may use methods that will let you solve these puzzles without using Triples. If so, you have my respect, but you don't have my cache. You have to find the Triples in these Sudokus to find the cache.

I fiddled with these diagrams and solved them six-ways-to-Sunday, but I may have missed something. It has happened before. If you notice a mistake or ambiguity, please just send me a gentle note and I'll get things fixed. Happy Puzzling, and thank you for doing my caches!