There's a few tricks to solving Kakuro which make things easier - we've got them all!

An empty square that has all its neighbouring squares (either column or row) filled in can easily be solved. Simply add together the corrsponding neighbouring values and then subtract the total from the clue. The remaining value is the answer for that square.

The lone square in the example therefore has to be 2 as 9-(4+3)=2

Find 2 intersecting runs and compare the clue number combinations for each. Any values which appear in the combinations for both runs are candidates for the square on which the runs intersect.

In this example, the "down" clue is 6, which has only 1 combination (1+2+3), whereas the "across" clue 20, has 4 combinations (3+8+9, 4+7+9, 5+6+9 and 5+7+8). However, the only common number in both sets of combinations is 3, therefore the intersection square must be 3.

This technique works by picking a run and performing a cross reference on every square along it, weening out combinations until you have only 1 left.

In this example we take the "across" run 12, which has the combinations (3+9, 4+8 and 5+7).

The first square (A) intersects a run with the combinations (1+4 and 2+3). Since neither 5 or 7 appear in these, we can discount the 5+7 combination from the first run. However, this still leaves us with the combinations (3+9 and 4+8).

The second square (B) intersects with a run which has only 1 combination (7+9), and since the (4+8) combination shares no common numbers we can remove it, leaving only 1 combination for our original run (3+9).

Knowing that this run has only 1 combination, we can now fill in this section of the puzzle.

In this example, we can deduce that the square outside of the blank 2x2 area shown must be a 3, we can do this using the filled area technique.

Firstly, add together all the "across" clues (4+3=7). Then add together all the "down" clues (4+6=10). Now work out the difference between those totals (10-7=3) and that will be the value of the square which is not in the 2x2 area.

Using this technique you can eliminate any number combinations which would lead to certain patterns of numbers (as shown in the examples).

The pattern shown the examples, with the numbers on the top row being the same as the bottom row (albeit in a different order), can never occur anywhere on a valid Kakuro board as it would lead to more than 1 solution.

If you swapped the top and bottom rows in the examples, the numbers would still add to the same clues, but would yield a different solution to the puzzle.

Bearing in mind that a proper Kakuro puzzle has only 1 solution, you should avoid any numbers which would create the patterns seen here.