This is a small Kakuro (Cross sums) puzzle which show how to use the technics described herearchive copy. This is just one order in which you can solve the puzzle.

This is the empty puzzle.

Rows and columns with only one possible combination of numbers.

The possible numbers for the sum 34 are 4, 6, 7, 8, and 9. For 7 they are 1, 2, and 4. The only one in common is 4.

The largest single number that can be found in a cell for the row whose sum is 9 is 6. That is because the other two cells require at least a 1 and a 2. Column 34 still has numbers 6, 7, 8, and 9 to be filled in, hence only 6 will fit in the given cell.

Only numbers 1 and 2 fit the top row and 1 and 3 fit the top-right column, therefore we place number 1 on the top right. Numbers 2 and 3 follow logically.

The row, whose sum is four, is still missing numbers 1 and 2. We know the correct order from two things:

a) Only 2 is large enough to sum up to 11.

b) Number 2 is required for the bottom right cell. Otherwise a sum can't be created for 19.