Teach Yourself Sudoku |

Learn the secrets that let you easily solve Sudoku puzzles faster! |

Lesson #8A. Check for TEAMS (Doubles – Triples – Quads - etc.) in each RCB
The numbers in any Sudoku work together in a variety of ways. Only one occurrence of 1 through 9 in each Row, Column, or Block. BLOCK OUTS and RESERVED Rows and Columns let you quickly deduce where other numbers can and can not go.
One of the most powerful ways that numbers work together in Sudoku is the concept of TEAMS. What’s a TEAM? It’s any group of numbers in an RCB that, in one way or another, share the same Cells (if that sounds slightly vague, I’ll explain it in detail as we move along.). The most common TEAMS are groups of 2, 3, 4, and 5 numbers but it’s possible to have teams all the way up to 9 numbers (although not very useful.)
Let’s start with a TEAM of 2 (also known as a Double). Check out the Sudoku puzzle fragment below.
By definition, a TEAM of 2 has: - has exactly a total of two different numbers in it (no more, no less) - the entire team “lives in” or occupies a total of two cells.
So lets look at the numbers here. We see that 2-3-4 occurs twice and lives in just two cells so that must be a TEAM of 2, right? Wrong. The TEAM 2-3-4 is made up of three numbers (2-3-4) so it can’t be a TEAM of 2. OK, how about this: Is 2-3-4 a TEAM of 3? Let’s see. It has three numbers (2-3-4), so it’s got that going for it. But wait, it only lives in two cells not three. Nope. The combination 2-3-4 doesn’t fit any of the rules we have for teams. It comes close but unfortunately that doesn’t count in Sudoku.
OK, what about the TEAM of 3-6. It looks like it could be a TEAM of 2. It has just two numbers (3-6) and the entire TEAM lives in a total of two cells. Guess what? That’s a TEAM of 2.
Yippee! A TEAM!
Uh, why should I care?
Here’s why. When you find a TEAM in an RCB of any size (2,3,4,5,etc.) you then KNOW with absolute certainty that the numbers on the teams MUST be placed in the TEAMS cells. Why is that important? It then allows you to erase any and all of the TEAMS number from any OTHER cells in the associated RCB. In effect, we’ve RESERVED the TEAM numbers exclusively for the TEAM cells. In our example, that means that even though the number 3 appears in two other cells (the 2-3-4 cells), we can remove it from those cells because the TEAM 3-6 has exclusively RESERVED other cells for the 3 (and for the 6 as well but the 6 doesn’t come into play in this example which is fine as the TEAM numbers may or may not be present in other cells.)
Here’s our updated Sudoku fragment. Notice how the 3’s have been removed from both the 2-3-4 cells.
Guess what? We now have a new TEAM in the Block, the 2-4 TEAM. In this example, it really doesn’t matter as there’s nothing else we can do with this small puzzle fragment but in real life, a good Sudoku player would immediately check the intersecting Rows and Columns for each of the 2-4 Cells to see if the 2-4 combination could be combined with any other cells or if the removal of the 3s created an ONLY ONCE situation in some other Cell in the associated Rows / Columns.
OK, so here’s an updated and more generalized version of our TEAM definition that we can use for TEAMS of any size, from 2 to 9 numbers (NOTE: I’ve included the theoretical maximum of 9 in this definition although it is essentially meaningless as a solving technique. In practice, most Sudoku puzzles, even the difficult ones, can usually be solved by looking for TEAMS of 5 or less. Just remember that larger TEAMS are valid and the same rules apply.)
A Sudoku TEAM of size X (X can be 2,3,4,5, etc.): - has a total of X different numbers in it (no more, no less) - the entire team “lives in” or occupies a total of X cells - for TEAMS of 3 or greater, it is not necessary for all of the TEAM members to live together in every cell although they may.
OK, everything was fine until that last one. I thought you said that TEAMS “live in” the same cells? Well, for a TEAM of 2, that’s true. But for 3 or more, it’s optional.
Let’s look at an example to help clarify this.
At first glance, there doesn’t seem to any TEAMS possible because all of the cells contain different numbers..
But take a close look at Cells containing 2-6, 2-6-7, and 6-7. While it’s true that that each of the cells has a different number combination in it (2-6 is not the same 2-6-7 is not the same as 6-7), it’s also true that each cell does have some combination of the three numbers 2-6-7 and together, they occupy a total of exactly three cells. Hmmm….three numbers sharing three cells. Maybe we’ve got something here.
Let’s go back and plug in some numbers in our TEAMS definition.
A Sudoku TEAM of size 3: - has a total of 3 different numbers in it (no more, no less) - the entire team “lives in” or occupies a total of 3 cells - for TEAMS of 3 or greater, it is not necessary for all of the TEAM members to live together in every cell although they may.
Check, check, and check. Well, now, looks like our 2-6-7 combination qualifies as a TEAM of 3. That means that we can go ahead and erase all of the other occurrences of 2-6-7 in the Block.
Here’s where the puzzle solving power of TEAMS really shines. Let’s remove 2-6-7 from all of the other cells that aren’t on the 2-6-7 TEAM. When we do, we find that: the 4-6 in the upper right cell becomes just a 4 (removed the 6) the 3-4-6 in the middle right cell becomes 3-4 (removed the 6) but that becomes just a 3 as we already have a 4 in the cell above the 2-3-5 in the lower right cell becomes 3-5 (removed the 2) but that becomes just a 5 as we already have a 3 in the cell above
Did you follow all that? Re-read it and take a look at the grid below and compare it to the previous one and work through the steps to make sure you understand what we just did.
Now will every TEAM always let you fill in three cells? No, not always but in any case, using TEAMS is such a powerful Sudoku solving technique that no matter what the immediate results, just the fact that you can remove one or more numbers from an Row, Column, or Block puts you that much closer to solving the puzzle. TEAMS are essential to solving almost all Sudoku so be sure to take your time and understand them thoroughly.
How do Rows / Columns / Blocks impact each other when using TEAM logic?
Our example was based on a Block and we were able to remove any other occurrences of 2-6-7 from that Block. What about the associated Row or Column? Can we remove 2-6-7 from those also?
To answer that we need to know where the TEAM lives. By that, I mean does it live in a Row, a Column, a Block, or a Combination?
Here are the new TEAM rules for erasing numbers:
Rule TB: If all of the cells for the entire TEAM reside in the same Block, you can remove / erase all other occurrences of the TEAM’s numbers from the other cells in that Block Rule TR: If all of the cells for the entire TEAM reside in the same Row, you can remove / erase all other occurrences of the TEAM’s numbers from the other cells in that Row Rule TC: If all of the cells for the entire TEAM reside in the same Column, you can remove / erase all other occurrences of the TEAM’s numbers from the other cells in that Column Rule TRB: If all of the cells for the entire TEAM reside in both the same Row and the same Block, you can remove other occurrences of the TEAM’s numbers from the other cells in both the Row and Block Rule TCB: If all of the cells for the entire TEAM reside in both the same Column and Block, you can remove all other occurrences of the TEAM’s numbers from the other cells in both the Column and the Block
Wow, that was a lot of verbiage. We’ll explore some examples and see how these rules play out in real life.
Let’s look again at the Sudoku fragment we’ve been using recently.
The TEAM of 2-6-7 is found in the first (2-6) and third (2-6-7, 6-7) rows so it doesn’t reside exclusively in any one Row. This means that we can’t remove 2-6-7 from either of the Rows (we’ll look at an example later where we can.)
The same can be said for the Columns. TEAM 2-6-7 lives in the first (2-6, 2-6-7) and second (6-7) columns so it doesn’t reside exclusively in any one Column.
So in this case, none of the other rules apply and we could only remove other occurrences of 2-6-7 from the Block which is exactly what we did using Rule TB listed above.
Now look at this slightly modified example.
We have the same numbers as in the original Block example but they’ve been rearranged slightly. The difference is that now TEAM 2-6-7 does live in the same Column (in this case, the first Column in the Block) in addition to living in the same Block. This gives us a double advantage because we now remove other occurrences of 2-6-7 from both the Block and the Column using Rule TCB.
NEXT UP: Lesson #8B. More TEAMS
Copyright 2006 Gary Ward All Rights Reserved |