SUMami

Introduced to 'Tough Puzzles' in 2004 after first being seen by us in the World Puzzle Championship of 2003, SUMami combines the logical techniques used in Hanjie and Kakuro.

The way this is done is that where a standard Hanjie puzzle uses a simple arrangement of black squares as the pattern to be discovered by the solver, in SUMami each black square is replaced by a single digit from 1 to 9. Instead of the clues providing the length of each block of squares, then, the SUMami clues provide the sum total of the digits in each block.

As in Kakuro, however, it is a rule that each block must contain no duplicated digits – and on top of this in fact, duplications are not permitted across an entire row or column.

These rules combined suggest some strategies to use – a first step is to note the minimum number of digits that could be used to form each block (9 is the largest number that could use just one square, 17 is the largest which could use just two, and so on) and armed with this knowledge use Hanjie logic to establish squares which must or cannot contain a digit. At this stage, it's not too important to try and work out which digit is used in any given square.

Approaching from another direction, it is equally useful to try and work out how many digits – and which – will be needed to fill each row and column. A row whose clues sum to 45 shows that all nine digits, and therefore nine squares, must be used; 44 or 43 show that eight digits must be used, omitting only the 1 or the 2 respectively.

By switching back and forth between the two logical techniques, the full solution will gradually emerge. Be aware, though, that unlike in Hanjie the pattern of digits in a SUMami does not form a picture.