I got many correct answers for this puzzle as well and a very good response. People who sent correct answers included Chandu Karan, Divya, Suman Saraf (The most complete solution) and Pratik Poddar. I am reproducing the solutions from Suman, done both by common logic and later set of all solutions generated from a computer script (I tried to clean them up for duplicates, but too many to do so :-()

Since we know the 4 corners will be counted twice, it means 55 (sum 1..10) + 4 corners should be a multiple of 4 since all sides all add up to the same sum. The sum of the corners could range between 10 (1,2,3,4 in the corners) to 34 (7,8,9,10) in the corners. This leaves us with the following options:

sum of sides | sum of corners

17 (*4 = 68) | 13

18 (*4 = 72) | 17

19 (*4 = 76) | 21

20 (*4 = 80) | 25

21 (*4 = 84) | 29

22 (*4 = 88) | 33

Using this option we can try and fill the frame. The last one is the simplest because there is only one combination that could fill the corners (i.e. sum of corners is 33) – 6,8,9,10. Using this we could fill it up in the following way:

6 7 9

5 3

1 2

10 4 8

Obviously a few transforms of this arrangement will also suffice.

Similarly, one can find solutions for sums = 18, 19 and 20

1 9 8

4 3

7 5

6 10 2

1 8 10

5 2

6 4

7 9 3

2 8 10

3 1

6 5

9 7 4

Exhaustive list of solutions:

results in:

sum(19)

1 8 10

5 2

6 4

7 9 3

sum(19)

1 8 10

5 2

7 3

6 9 4

sum(18)

1 9 8

4 3

7 5

6 10 2

sum(19)

1 10 8

4 2

5 6

9 7 3

sum(20)

2 8 10

3 1

6 5

9 7 4

sum(18)

2 10 6

3 1

8 7

5 9 4

sum(19)

3 9 7

1 2

10 4

5 8 6

sum(20)

3 9 8

2 1

5 7

10 6 4

sum(22)

6 7 9

1 2

5 3

10 4 8

sum(20)

8 9 3

1 2

7 5

4 6 10

sum(20)

8 9 3

1 5

7 2

4 6 10

sum(20)

8 9 3

7 2

1 5

4 6 10

sum(20)

8 9 3

7 5

1 2

4 6 10

sum(19)

8 10 1

2 4

6 5

3 7 9

sum(19)

8 10 1

2 5

6 4

3 7 9

sum(19)

8 10 1

6 4

2 5

3 7 9

sum(19)

8 10 1

6 5

2 4

3 7 9

sum(19)

9 7 3

4 2

5 6

1 10 8

sum(19)

9 7 3

4 6

5 2

1 10 8

sum(19)

9 7 3

5 2

4 6

1 10 8

sum(19)

9 7 3

5 6

4 2

1 10 8

sum(20)

9 7 4

3 1

6 5

2 8 10

sum(20)

9 7 4

3 5

6 1

2 8 10

sum(20)

9 7 4

6 1

3 5

2 8 10

sum(20)

9 7 4

6 5

3 1

2 8 10

sum(22)

9 7 6

1 3

2 5

10 4 8

sum(22)

9 7 6

1 5

2 3

10 4 8

sum(22)

9 7 6

2 1

3 5

8 4 10

sum(22)

9 7 6

2 3

1 5

10 4 8

sum(22)

9 7 6

2 5

1 3

10 4 8

sum(22)

9 7 6

2 5

3 1

8 4 10

sum(22)

9 7 6

3 1

2 5

8 4 10

sum(22)

9 7 6

3 5

2 1

8 4 10

sum(22)

10 4 8

1 2

5 3

6 7 9

sum(22)

10 4 8

1 3

2 5

9 7 6

sum(22)

10 4 8

1 3

5 2

6 7 9

sum(22)

10 4 8

1 5

2 3

9 7 6

sum(22)

10 4 8

2 3

1 5

9 7 6

sum(22)

10 4 8

2 5

1 3

9 7 6

sum(22)

10 4 8

5 2

1 3

6 7 9

sum(22)

10 4 8

5 3

1 2

6 7 9

sum(20)

10 6 4

2 1

5 7

3 9 8

sum(20)

10 6 4

2 7

5 1

3 9 8

sum(20)

10 6 4

5 1

2 7

3 9 8

sum(20)

10 6 4

5 7

2 1

3 9 8

sum(19)

10 8 1

2 5

3 7

4 9 6

sum(19)

10 8 1

2 5

4 6

3 9 7

sum(19)

10 8 1

2 6

4 5

3 9 7

sum(19)

10 8 1

2 7

3 5

4 9 6

sum(19)

10 8 1

3 5

2 7

4 9 6

sum(19)

10 8 1

3 7

2 5

4 9 6

sum(19)

10 8 1

4 5

2 6

3 9 7

sum(19)

10 8 1

4 6

2 5

3 9 7

sum(20)

10 8 2

1 3

5 6

4 7 9

sum(20)

10 8 2

1 6

5 3

4 7 9

sum(20)

10 8 2

5 3

1 6

4 7 9

sum(20)

10 8 2

5 6

1 3

4 7 9

Hope you all enjoyed the puzzle!