Groups / Sets

## Definition

The upcoming techniques (Hidden Triples, Hidden Quadruples, Naked Triples, Naked Quadruples) are all looking for so-called Sets/Groups in the Sudoku grid, so let us dive into what they are and how they are used.

This section gets a bit theoretical, but bear with me, for using this information can greatly increase your Sudoku solving efficiency, and is necessary to fully understand some techniques.

Let us start by how they could be defined.

A group oftentimes means a group of cells within a region.

A set generally refers to a set of candidates (here: 1, 4, 6).

Combine the two and you have a so called Locked Set or Locked Group (depending on the viewpoint).

Here, exactly three candidates are locked in place in a group of exactly three cells.

I have decided to use the term "Groups" synonymously with "Locked Sets / Groups", since it needs to be frequently used, and since this website is for a general audience.

In case I mean a group of cells in general, I will explicitly say "group of cells".

So:

• A 1-Group is one candidate pinned down to exactly one cell within a region.
• A 2-Group is two candidates pinned down to exactly two cells within a region.
• A 3-Group is three candidates pinned down to exactly three cells within a region.
• ...
• A 9-Group is nine candidates pinned down to exactly nine cells within a region.

## 1-Group

A 1-Group is basically just a Single.

(It does not matter how this Single came to be. It could be a given. It could be a Hidden Single that you have discovered, it could be a Naked Single.)

A Single (being one candidate pinned down to one cell) is always a cell's solution, so the candidate 4 from the previous example is nothing else than a big 4.

We can write this 1-Group as "(4)-Group", i.e. a group with only the candidate 4.

## 9-Group

The other extreme is the 9-Group, where nine candidates are pinned down to nine cells.

A 9-Group gives us no information, whatsoever. (Any number can go in any cell.)

We can write this 9-Group as (1,2,3,4,5,6,7,8,9)-Group, i.e. a group with only the candidates 1, 2, 3, 4, 5, 6, 7, 8, 9.

## Splitting a group

A group can be split into smaller groups.

The easiest example for this is you finding a Single.

Imagine you find out that one of those cells is a 3.

You have now created two groups. The (1,2,3,4,5,6,7,8,9)-Group becomes a (1,2,4,5,6,7,8,9)-Group and a (3)-Group.

You finding a pair (e.g. a Hidden Pair) would also further split the remaining groups.

We are now down to three groups:

• a (2,4,5,6,8,9)-Group ()
• a (3)-Group ()
• a (1,7)-Group ()

## Groups are disjoint

Groups are always disjoint, i.e. no two groups within a region share a candidate.

So this example shows only one group, not two, not three.

(Because you cannot divide that 9-Group into smaller groups, so that the candidates are not somehow shared between the resulting groups.)

If you, however, find one group within another group, you can always be sure, that the remaining cells are now also a group themselves.

Take this grid for example. Imagine you have just found out that there are three cells that can only be the three candidates 2, 4 and 6.

You now immediately know that the rest of the former group where the 2, 4 and 6 were in, now also form a group. A group, in which 2, 4 and 6 are not allowed.

This last example is the main reason why it is so important to know about groups.

You finding one group always result in another group forming where the candidates of the first group are eliminated.

## Groups with fewer candidates

As seen in the last example, groups don't have to always have each candidate in every cell of the group.

So it will not always be like in this example.

In actual Sudokus, the majority of groups will look like in this example.

Even though not all cells of the group have all three candidates in them, they are still a group.

So we immediately know that the other cells in that row will not hold 1s, 2s or 3s. (And neither do the other cells of that box.)

## A real life example

Here is a Sudoku position, where we can find a pair (2-Group).

We know that the candidates from this group cannot be in another cell within that region (box), so we eliminate the 7s.

This is a Naked Pair, by the way.

This results in another pair in that region: a (1,4)-Pair.

## Why 5-Groups and higher are irrelevant

In theory there are even 9-Groups, but it does not make sense to look for any group bigger than a 4-Group (Quadruple). (That applies to Hidden Groups, as well as Naked Groups.)

This example shows why. (Please only look at the rows. Columns and boxes do not exist here.)

Every row, column and box can hold exactly nine numbers. If you have pinned down two candidates to exactly two cells, then you know that the other seven candidates must be in the other seven cells.

This example shows that for differently sized groups within the rows.

So if you were able to find a 5-Group, then you could also find the complementary 4-Group. If you were able to find a 6-Group, you could also find the complementary 3-Group, etc.

This, of course, is only true in 9x9 Sudokus. In bigger Sudokus, bigger groups make sense again.

## How Naked Groups and Hidden Groups are connected

• The Naked Group technique requires you to find a Naked Group and reveals the complementary Hidden Group.
• The Hidden Group technique requires you to find a Hidden Group, making it a Naked Group.

Take this example.

The red cells are a Naked Pair.

They can only be 8 or 9, because they both see all other numbers.

The other two empty cells of the region are the Hidden Pair. 3 and 4 are forbidden in all other cells, so they need to be in the remaining two.

So the splitting of the (3,4,8,9)-Group (the four empty cells)...

... result in two groups. One was hidden ((3,4)-Group) and one was naked ((8,9)-Group).

## How to solve a Sudoku

With the knowledge of groups, we have now another way of thinking about how a Sudoku is solved:

The goal is basically to split all groups (in all regions) into smaller and smaller groups, until there are only 1-Groups left.

(As always: the more perspectives you have on anything, the better.)