Almost Locked Sets

In almost every puzzle you can find the almost locked sets. But what does 'almost locked set' mean?

A locked set is a set (a set is a collection of numbers of cells which are in the same row, column or box) where the number of candidates matches the number of cells they are in. We say the set is locked, because we know all the possible candidates for the cells but we do not know which number goes to which cell. An example of a locked set can be found in the example. The cells A2 and C1 are a locked set. Both cells contain the same numbers. So the final solution of each cell is one of those numbers.

A almost closed set is a locked set with one extra candidate. The cells C1 and C9 are an example for a almost closed set. The total amount of possible solutions (candidates) is 3 en the total amount of cells is two. So there is one candidate extra.

The XZ-rule :

Almost Locked Set example 1

Almost locked sets appear very often, but are not always useful. To make use of Almost Locked Sets you will need at least two of them. The size of the sets is not important, but the sets have to be able to 'see' each other. This means they have to share a row, column or box.

They also need to have a common candidate. If the common candidate is among the cells that can 'see' each other' then this candidate can not exists in both sets. We call this candidate a restricted candidate. In the example below the number 6 would be a restricted candidate. This candidate is also called the X-candidate.

The Z-candidate can be any other candidate which is found in both sets and can exist in both sets at the same time. An example of a Z-candidate is the number 8 in the example on the left.

The general rule is that any number which is equal to a Z-candidate can be removed if and only if this number can 'see' all the Z-candidates in both sets.

Is This true? Yes, it is. Just try it. Suppose cell E1 has the number 8 as the solution. This means that in cell C1 the number 4 the solution is and in cell C9 number 6. Therefore the number 6 and 8 can be removed from cell E9, leaving number 1 as the solution. This means that number 1 can be eliminated from cell E2 and also number 8 (8 is in cell E1). This will leave cell E2 empty without any candidates, which is not possible. Therefore number 8 can not be the solution for cell E1 and can be removed from this cell.

Almost locked set with one cell and three cells:

Almost Locked Set Example 2

An Almost Closed Set can also contain just one cell. This cell should then contain 1+1 candidates. In the example left the cell A6 is a Almost Locked Set with one cell. A second set is formed by the cells D3, D5 and D6. The set of candidates in the last set are the numbers 1,2,3 and 9. This are four numbers for three cells, which makes this set a Almost Locked Set.

The number 1 is the restricted candidate (can not exist in both sets at the same time). The number 3 is the Z candidate, because the number 3 in cell A6 can not 'see' all the other candidates number 3 in the green set.

The number 3 in cell E6 can see all the Z-candidates in both sets and can be removed according to the XZ-rule.

More complex examples

Almost Locked Set Example 3

In this example we see a large Almost Locked Set comprising of 5 cells en one small set, comprising of 1 cell. The number 5 appears in both sets one time. Both numbers 5 can 'see' each other, which makes them a restricted candidate.

The number 7 in cell E9 is able to 'see' all other number 7 in both set and can according to the XZ-rule be removed.

Almost Locked Set Example 4

Here we see an Almost Locked Set with two cells and an Almost Locked Set with 4 cells. The number 7 appears in both sets, but not all number 7 can see each other. This make number 7 the Z-candidate. The number 7 in cell G7 can 'see' all the numbers 7 in both sets and can be eliminated.