Standard Sudoku rules. Place numbers from 1 to 9 in every row, column, and 3×3 box.
Additionally, outside the grid, there will be some numbers. The numbers indicate the sum of the numbers physically between 1 and 9 in the respective row or column. (that is, if the row is 953126847, the sum of the numbers between 1 and 9 is 8)
In the first 5 column, the only solution can be 1-2-3-9 going downward.
In the 31 column, only a 4 goes outside the 1-9 group. The 9 can't be at the top, because the 1 won't go next to another 1. There is already a 9 in the second row. There is already a 1 in the top row, so it must be 1 in second row, 9 in bottom row.
In the 4 column, the only solution can be 1-4-9 going upward.
In the 9 column, the only solution can be 1-5-4-9 going downward.
In the 3 column, the only solution is 9-3-1 going downward.
In the other 5 column, the solution can't be 1-5-9 downward because there is already a 9 in the bottom row. It can't be 1-2-3-9 or 1-3-2-9 going upward because there is already a 9 in that row. So it must be 1-5-9 going upward.
In the bottom right block, the 9 can only go in the 8 column. In the 8 column, there is only one place for the 1 to go, for 1-2-6-9 or 1-3-5-9. The only combination that fits is 1-6-2-9.
I then continued with the normal Sudoku solving methods. Later on the 15 clue was used to find a unique pattern, and the 26 clue wasn't needed - but verified later.