- Fill in the single square cages.
- Look for gimmes like 3+ (the cage always consists of two squares). This can only be 1/2 or 2/1. I put a 2 in the top of the left (or top) square and a 1 in the top of the right (or bottom) square. Then I put the numbers in the reverse order in the bottom of the squares. At this point we don't know the final ordering of the numbers, but, and this is important, we know that 1 and 2, that's two of the six digits in the row or column, are spoken for, so that later on, when we are working in this row or column, we need only consider the digits 3, 4, 5, and 6.
- Same for 4+ in two squares. 3 and 1 are now used up.
- Repeat the first step for 11+ (only if the cage is two squares), this time though with the digits 5 and 6. In this case the remaining digits in the row or column can only be 1, 2, 3, and 4.
- Same for 5- (all minus cages consist of two squares), taking up 1 and 6, and leaving 2, 3, 4, and 5 for the remaining digits in that row or column.
- 25x is a special target number. It can only appear in cages with more than 2-squares and with 5s in two of the squares. 3-square cages are completely determined and higher square cages are undetermined. 125x, with three 5s, is legal, but I've never seen it (in a 6 X 6 puzzle).
- Remembering that times cages can have any number of 1s, look for either of the primes 3 or 5 as a target number, that is 3x or 5x. If the cage is two squares, you can proceed to pencil in 3/1 or 5/1 as we did in step one above.
- Look for any times cages with a target number ending in 0 or 5. Those targets have a (prime) factor of 5, which, say, unlike 4 or 6, can't be formed by products of smaller numbers (other than 1). Also, unlike the prime 3, 5 can't be combined with another digit to form a larger number. 3 can masquerade as 6, but 5 can't masquerade as 10 (too big). These times cages have a 5 in one square. Pencil a 5 in each square. If it is a 2-square cage, you can actually pencil in two numbers, e.g., 20x is 5/4. Cages with more than 2 squares require some thought. For example, 30x could be 5/2/3 or 5/1/6. I'd fill in only a 5 in the squares, but remember that the remaining squares can only be 2/3 or 1/6. For 2-square cages the 5 and another digit is spoken for in that row or column.
- Next are 3/ (3 divides). There are two pairs, or four possbilibities, 6/2 and 3/1. You can pencil in the four possibilities, but maybe some are already eliminated by other entries. Check for that.
- 2/ have three possible pairs, 6/3, 4/2, and 2/1, which means 6 possibilities. Check for any that you can eliminate.
- For cages with multiplicative targets it often helps to write out the (unique) prime factorization of the target. There can only be three different prime factors, 2, 3, and 5. For example, you will never see 14x. If the cage consists of two squares, the factorization may determine the two entries, e.g., 8 = 2 * 2 * 2, so the entries have to be 4 and 2, in some order. However, 12x (2 * 2 * 3) in two squares could be either 3 and 4 or 2 and 6. 12x in three squares has differing possibilities depending on whether the squares are colinear or not. If they are colinear, you can't have 2, 2, and 3, but those entries are OK if you have non-colinear squares so that you can put the two 2's in different rows (or columns). And three (or more) squares gives room for 1s, as in 1, 3, and 4, or 1, 2, and 6.
- An odd multiplicative target, e.g., 45x, can't have a 6 in it. It could have a 3, however, as in 45x.
- If an odd multiplicative target is not divisible by 3 (check by adding digits or test dividing by 3), e.g., 25x, it can't have either a 3 or 6, and if the cage is colinear, that eliminates two choices from the remaining entries of that row or column.
- Sometimes, you can quickly determine the numbers that are
**not**in a cage, narrowing the possibilities for the rest of the row or column. The target 4- cannot have a 3 or 4 in either of its squares. - Multiplicative targets without a 5 in their prime factorization, say 36, eliminate the digit 5 from all of their squares. Likewise, multiplicative targets without a 3 in their prime factorization, say 40, eliminate both 3 and 6 from their squares. And multiplicative targets without a 2 in their prime factorization, say 45, cannot have a 2, 4, or 6 in their makeup.
- Any mutiplicative target with a single 2 in it's prime factorization, cannot have a 4 in any square. To have a 4 requires not just an even target, but one with at least 2 2s in its factorization, like 20.