Each cell in a grid contains one of the digits from 1 to 7. Each row and column has each digit exactly once. The following clues give the totals of the cells indicated. How can you fill the grid?

A56=13

ABC1=6

ABCD4=13

B234=18

BC7=5

C34=4

D67=13

DEF2=8

EFG3=16

EF5=6

FG5=4

G1234=17

I made my grid with rows A-G from top to bottom and columns 1-7 from left to right.

1.) Given that EF5=6 and FG5=3, E5 has to be 5, F5 has to be 1, and G3 has to be 3.

2.) Since C34=4, the numbers must be 1,3 or 3,1. Since ABC1=6 and
BC7=5, then C1 must=2 and C7 must = 4. So obviously, B7=1, B1=3 and
A1=1.

3.) EFG3=16. From this, all possible number combinations, from top to
bottom are: 3,6,7 | 3,7,6 | 4,5,7 | 4,7,5 | 5,4,7 | 5,7,4 | 6,3,7 and
6,7,3 and 7,3,6 The two 5 combinations as well as the final one can be
eliminated because of the 5,1,3 directly to the right.

4.) C3 must be either 1 or 3. If it is three, the grid does not work
out about 3/4 of the way through. I ended up with 2, 3's in one row. So
it has to be 1, which means C4=3.

5.) B234=18 This means the numbers must be 5,6 and 7. In any
combination, if B4= 5,6 or 7, one of the remaining squares must= 1 in
order for ABCD4=13. So D4 must be 1 in any case.

6.) E2 must be 1 becuase 1,2,5 and 1,3,4 are the only possible
cobinations of 8. Both have a 1 and E2 is the only possible spot.

7.) For DEF to equal 8, there are 4 possible combinations with E2 being
1. 4,1,3 | 3,1,4 | 5,1,2 | and 2,1,5. None work out except 3,1,4.

8.) From here, it is easy. D1 must be 5, G1 must be 4 because the combinations for 6 and 7 do not work out.

I got:

1542763

3657241

2713654

5321476

6174532

7436125

4265317

Nice puzzle :)