Can you solve this kakuro puzzle?

Question

This kakuro puzzle comes from the newest edition of Topple, an indie puzzle magazine I run. For those who are unfamiliar with kakuro, here are some instructions:

Kakuro consists of a blank grid with sum-clues in various places. The object is to fill all empty squares using numbers 1 to 9 so the sum of each horizontal block equals the clue on its left, and the sum of each vertical block equals the clue on its top. In addition, no number may be used in the same block more than once.

Answer 1

The final solution is:

Here are the steps:

First of all, focus on the top-left corner:

Here the only possibility for the vertical 23-run is 9 8 6. But 9 or 8 in d would render the 12-run unsolvable, so d=6. Then a+b+c=6, so they are 1,2,3 in some order. But there's no possible combination for 27 with 1 or 2, so a=3. Then the 27-run must be 3 7 8 9 in some order; but g can't be 9 or 8, so g=7, f=1; so b$\ne$1$\implies $b=2,c=1,e=2. Since the horizontal 23-run is 9 8 6, and we need 12 more for the 15-run, the only possibilities for the rest of the 15-run are 4 8 or 3 9. But in the first case, we'd need another 16 for the 22-run, forcing a 7 between a and g, which is bad since g=7. So the part is something like:

Now we'll solve four very similar regions. Focus on this part:

Clearly the only possibilities for the 12-run are 7 5 or 8 4. But the first case would imply d=1, b=7=a, which is bad. So (a,b,c,d)=(8,6,2,4). Similar strategy can be applied to the other 2x2 parts, giving:

Now that we know the 23-run considered in the first paragraph is 6 8 9, we can do some usual sum-chasing to arrive at the following:

Now,

we apply some reasoning along the line "S has the only combination a1 a2 a3, but that cell can't have a1 or a2, so it's a3", to arrive at the following:

Next, consider the top-right corner.

The only possibility for the vertical 24-run is 9 8 7 in some order, and for 30, it's 9 8 7 6 in some order. So their intersection must be 9 or 8. If it's 9, then the 16-run must be 7 9, and the 24-run would be 9 8, which means we need to complete the 19 run from 8, 9 and 2 more numbers, which is impossible. So the only possibility is: 8 9 * * and 9 7 * *. Also, since the last two asterisks are 1 2 in some order, and there's no combination for 4-cell 26-run with a 1, we get that these two rows are 8 9 7 6 and 9 7 2 1. Now some sum-chasing leads to this:

Now let's try the bottom-left corner.

The horizontal 23-run has to be 9 6 8, or 6 9 8; but the first possibility would mean we need to get a sum of 4 from 3 cells in the 13-run, so it has to be 6 9 8. The other three cells in the 13-run have to be 1 2 4 in some order. How 1 or 2 in that 12 run isn't possible, so the next row is 4 8.

Now focus on the remaining 2x4 grid. By summing by rows and then by column, we conclude that the yellow cells must add up to 13; but the only possible combination for the 11-run is 1 2 3 5 in some order, so these yellow cells must be 5 8. Now the remaining cells in the bottom row are 1 2 4 in some order, and 4 fits only in the second cell. Now sum chasing yields the following:

To finish things off, look at the bottom right:

it's simple to get that the 23-run is 6 8 9 and the 7-run begins with 4. Now look at the 22-run. It's first element is 1 or 2 and the third element is among 1 2 3. So the other two cells must sum up to at least 22-(2+3)=17; but that's the maximum for two cells, so equality holds, and the 22-run is precisely 2 9 3 8. Now some quick casework yields the following:

Now it's solved completely!