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Innies and Outies

The innies and outies technique is a very powerful way of determining the sum of a cell, from the sums of several cages. It relies on the observation that every group (row, column or box) must contain all the digits from 1 to 9, meaning that every group must have a sum of 45. If, therefore, we are able to determine the sum of eight cells within a group, we know what digit must go in the leftover cell. This is an "Innie". If, on the other hand, we can determine the sum of 10 cells, 9 of which are contained in one group, we know what digit must go in the cell which is outside the group. This is an outie.

Here is an example of an innie:

The innie technique allows us to place the digit "7" in the cell highlighted in red. We know that the highlighted group must have a sum of 45. We also know that the cells highlighted in orange all belong to two cages, whose sums are 14 and 24. This means that the sum of the orange cells must be 38. The difference between 45 and 38 is 7, so the digit "7" must go in the remaining cell.

Here is an example of the outie technique:

The outie technique allows us to place a "4" in the cell highlighted in red. We know that the cells highlighted in orange must have a sum of 45 (because they form a complete box). We also know that the sums of the 3 highlighted cages are 11 + 18 + 20 = 49. The difference between 49 and 45 is 4, so we can put a "4" in the red cell.

Sometimes, innies and outies can appear as the puzzle progresses. For instance:

In this example, we can place the digit "2" in the cell highlighted in red. The highlighted box would not normally form a useful innie in this way, but because we have filled in two cells, we can determine the sum of the cells in orange: 14 and 18 for the two cages, and 5 and 6 for the two filled cells. 14 + 18 + 5 + 6 = 43, which means that the cell highlighted in red must be a "2".