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## Killer Cage Combination ExampleFrom sudokuwiki.org, the puzzle solver's site |

This example shows the kind of logic involved in using cage combinations in Killer Sudoku. There are many leaps of logic to be found in this type of puzzle, and this should also explain in more detail some of the results from the Killer Sudoku Solver. In this instance the solver returns

KILLER STEP (Hard) on D3: shape of length 3 with clue of 15+ can only be 1/2/3/4/5/6/8/9, removing 7

The question is - why does it eliminate the 7 in D3 and not D1?

Hovering over the 15 on the small board it lists the combinations. There are quite a few in this cage:

1/5/9

1/6/8

2/4/9

2/5/8

2/6/7

3/4/8

3/5/7

4/5/6

KILLER STEP (Hard) on D3: shape of length 3 with clue of 15+ can only be 1/2/3/4/5/6/8/9, removing 7

The question is - why does it eliminate the 7 in D3 and not D1?

Hovering over the 15 on the small board it lists the combinations. There are quite a few in this cage:

1/5/9

1/6/8

2/4/9

2/5/8

2/6/7

3/4/8

3/5/7

4/5/6

Now...

1/5/9 <-- no 7 used at all

1/6/8 <-- no 7 used at all

2/4/9 <-- no 7 used at all

2/5/8 <-- no 7 used at all

2/6/7

3/4/8 <-- no 7 used at all

3/5/7

4/5/6 <-- no 7 used at all

So if 7 is used (either D1 or D3) the combos are

3/5/7

2/6/7

But there is no 3 in the cage, so we are left with

2/6/7

Try and fit that in the cage - it can only go one way with the candidates that remain:

7/6/2 fits in D1,D2,D3

But there is no 2 and 6 in D1 and D2 if D3 is a 7

Voila...if - and its still and IF...7 is used the solution will be {7,6,2}, so the solver will remove 7 from D3.

Of course we don't know if 7 is actually used in the combo so that's as far as we can take it at this stage.

1/5/9 <-- no 7 used at all

1/6/8 <-- no 7 used at all

2/4/9 <-- no 7 used at all

2/5/8 <-- no 7 used at all

2/6/7

3/4/8 <-- no 7 used at all

3/5/7

4/5/6 <-- no 7 used at all

So if 7 is used (either D1 or D3) the combos are

3/5/7

2/6/7

But there is no 3 in the cage, so we are left with

2/6/7

Try and fit that in the cage - it can only go one way with the candidates that remain:

7/6/2 fits in D1,D2,D3

But there is no 2 and 6 in D1 and D2 if D3 is a 7

Voila...if - and its still and IF...7 is used the solution will be {7,6,2}, so the solver will remove 7 from D3.

Of course we don't know if 7 is actually used in the combo so that's as far as we can take it at this stage.