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 is9 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 be3 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 is9 8 6
, and we need 12 more for the 15-run, the only possibilities for the rest of the 15-run are4 8
or3 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 are7 5
or8 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 is6 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 combinationa1 a2 a3
, but that cell can't havea1
ora2
, so it'sa3
", 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's9 8 7 6
in some order. So their intersection must be 9 or 8. If it's 9, then the 16-run must be7 9
, and the 24-run would be9 8
, which means we need to complete the19
run from 8, 9 and 2 more numbers, which is impossible. So the only possibility is:8 9 * *
and9 7 * *
. Also, since the last two asterisks are1 2
in some order, and there's no combination for 4-cell 26-run with a 1, we get that these two rows are8 9 7 6
and9 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
, or6 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 be6 9 8
. The other three cells in the 13-run have to be1 2 4
in some order. How1
or2
in that 12 run isn't possible, so the next row is4 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 is1 2 3 5
in some order, so these yellow cells must be5 8
. Now the remaining cells in the bottom row are1 2 4
in some order, and4
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 with4
. Now look at the 22-run. It's first element is1
or2
and the third element is among1 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 precisely2 9 3 8
. Now some quick casework yields the following:
Now it's solved completely!