i didn`t use any methods in the beginning, but when i found the first 1,4,9 i saw that it was exactly the same numbers as in the chess square puzzle I just added the extra numbers,
Original chess puzzle:
1 + 4 + 9 + 16 + 25 + 36 + 49 + 64=204
3 - 5- 7 - 9 - 11 - 13- 15 difference in the numbers in the line over
2 - 2 - 2 - 2 - 2 - 2 difference in the numbers in the line over

With extra numbers:
1 – 4 – 9 – 16 – 25 – 36 – 49 – 64 – 81 – 100
3 – 5 – 7 – 9 – 11 – 13 – 15 – 17 - 19 difference in the numbers in the line over
2 – 2 – 2 – 2 – 2 – 2 – 2 – 2 – difference in the numbers in the line over

The lucky prisoners are I cell`s: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Hope this is somehow understandable

i didn`t use any methods in the beginning, but when i found the first 1,4,9 i saw that it was exactly the same numbers as in the chess square puzzle I just added the extra numbers,

Original chess puzzle:

1 + 4 + 9 + 16 + 25 + 36 + 49 + 64=204

3 - 5- 7 - 9 - 11 - 13- 15 difference in the numbers in the line over

2 - 2 - 2 - 2 - 2 - 2 difference in the numbers in the line over

With extra numbers:

1 – 4 – 9 – 16 – 25 – 36 – 49 – 64 – 81 – 100

3 – 5 – 7 – 9 – 11 – 13 – 15 – 17 - 19 difference in the numbers in the line over

2 – 2 – 2 – 2 – 2 – 2 – 2 – 2 – difference in the numbers in the line over

The lucky prisoners are I cell`s: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Hope this is somehow understandable