cage totaling 13 has the possible combinations of (1, 2, 3, 7), (1, 2, 4, 6), or (1, 3, 4, 5). Even though, initially, there is no way to tell which combination of numbers is correct, every solution available has a 1 in it. The player then knows for certain that one of the numbers within that cage is 1 (no matter which is the final solution). This can be useful if, for example, they have already deduced another cell within a nonet the cage resides in as having the number 1 as its solution. They then know that the 1 can only reside in cells that are outside of this nonet. If there is only one cell available, it is a 1. Initial analysis of the sample problem Fewest possible combinations The two cells in the top left must be 1+2. The 3 cells to the right totaling 15 cannot therefore have either a 1 or a 2, so they must be either 3+4+8, 3+5+7, or 4+5+6. The two vertical cells in the top left of the top right nonet cannot be 2+2 as that would mean duplicates, so they must be 1+3. The 1 cannot be in the top line as that conflicts with our first 2 cells therefore the top cell of this pair is 3 and the lower cell 1. This also means the 3 cell cage 15 to the left cannot contain a 3 and so is 4+5+6. Similarly the neighbouring 16 must be 9+7. The four cells in the top right cage (totaling 15) can only include one of 1, 3, 7, or 9 (if at all) because of the presence of 1, 3, 7, and 9 in the top right hand nonet. If any one of 1, 3, 7, or 9 is present then this must be the lone square in the nonet below. Therefore, these 4 cells are one of 1+2+4+8 or 2+3+4+6. The 2 cells in the middle of the left edge must be either 1+5 or 2+4; and so on. 45 rule example Looking at the nonet on the left hand side in the middle, we can see that there are three cages which do not cross over into another nonet; these add up to 33, meaning that the sum of the remaining two cells must be 12. This does not seem particularly useful, but consider that the cell in