From Sudopedia, the free Sudoku reference guide

From Sudopedia

Locked Cages is an intermediate Killer Sudoku solving technique that, in its simplest (unoptimized) form, can be expressed as follows:

"If digit X in house H is locked within 2 or more cages, all of which have the property that, if they contain the digit X, they must also contain the digit Z, then candidate Z can be eliminated from all common peers (if any) of all candidate positions for digit Z in all cages."

The above principle is based on the fact that one of the cages concerned must contain the digit X in House H, and therefore must also contain the digit Z. If a cell can see all candidate positions for digit Z for all of these cages then (regardless of which cage contains the digit X in house H) it cannot itself contain the digit Z.

Optimization

If any of the cages concerned contains only one candidate position for digit X within house H, then this cell can be excluded from the list of candidate positions for digit Z when applying the above rule, since (if this cage contains the digit X of house H) this cell must contain the digit X.

Example 1 (Simple case)

The following example is taken from Assassin 45. Note that the digit 9 in column 9 is locked within the two 11(2) cages at r12c9 and r89c9. If either cage contains a 9, it must also contain a 2. Hence, one of the two cages must be {29}. The digit 2 is therefore also locked within r1289c9, allowing the candidate 2 to be eliminated elsewhere in column 9 (r345c9).

Example 2 (Alternative cage orientation)

The following example is taken from Maverick 1. The digit 4 in column 7 is locked within the two 12(2) cages at r2c78 and r8c78. One of these cages must be {48}, implying that the digit 8 in column 8 must be locked within r28c8. The candidate 8 can thus be eliminated elsewhere in column 8 (r4569c8).

Example 3 (Example with a 3-cell cage)

The following example is taken from Assassin 2X. The digit 6 in row 5 is locked within the 10(3) cage at r5c123 and the split 7(2) cage at r5c5+r6c4. Thus, either the 10(3) cage is {136}, or the split 7(2) cage is {16}. The digit 1 is therefore locked within r5c12+r6c4, allowing the candidate 1 be eliminated from the common peer cells at r6c123.

Example 4 (More complicated scenario)

The following example is taken from the Vortex Killer. The digit 8 in nonet 2 (N2) is locked within the complex (innie/outie difference) cage at r1c6+r2c4+r4c5 and the 11(3) cage at r2c56+r3c6. The two N2 innies at r1c6+r2c4 = r4c5 + 9. The possible permutations for the complex cage at r1c6+r2c4+r4c5 are thus [641/652/821/832]. The possible combinations for the 11(3) cage at r2c56+r3c6 are {128/146/245}. Thus, if either cage contains the digit 8, it must also contain the digit 2. Since one of these 2 cages must contain the digit 8, the digit 2 is locked within r2c456+r3c6+r4c5. Therefore, the candidate 2 can be eliminated from the common peer cells at r3c45.

Notes

All 2-cell cages concerned must have the same cage sum (X + Z).