## Kakuro Puzzles

### What is Kakuro? Kakuro is a kind of arithmetic cross-word, where the clues are sums and the answers are sequences of digits. Kakuro is played on a typical crossword grid in which each cell is part of two clues: an across clue and a down clue. The rules are simple: fill in the empty cells with single digits from 1 to 9 such that each horizontal and vertical block adds up to the sum shown on the block. You must not repeat any digit in a block.

Although kakuro puzzles look as though they demand a lot of arithmetic, they are really logic puzzles. The intersections of across sums and down sums and the restriction to unique single digits in each cell of a block creates a puzzle that has many similarities to sudoku. Solving a kakuro puzzle, like solving a sudoku puzzle, involves finding a sequence of logical deductions that takes you from the initial setting to a unique solution.

Kakuro puzzles can be of any size, from about 5x5 for the smallest interesting puzzle, to giant puzzles that are 30 by 30. A typical size is around 9-15 cells square. Kakuro puzzles originated in the US about 50 years ago under the name "cross-sum" puzzles. In Japan they have become known as 'kakro' puzzles, and in the U.K. we have anglicised that name to get kakuro.

### Getting started with kakuro

The key to solving kakuro puzzles is to be familiar with 'kakuro blocks' - a list of special number clues. These are blocks of a given size and sum that only have a solution in terms of a single choice of digits. For example, the sum of 16 on a block of length two can only be made with the digits 7 and 9. Thus wherever you see this sum on a block this size on a puzzle, you know immediately that the only digits that can be placed in the cells are a 7 and a 9, although you may not know which way around they go. Similarly a sum of 24 on a block of length three can only be made with 9, 8 & 7.

This table gives you all the kakuro blocks between 2 and 7 cells in length (blocks of length 1, 8 and 9 are somewhat easier to deal with, can you see why?).

Block lengthSumUnique digits Unique digitsSumBlock length
231+2 3+4+5+6+7+8+9427
41+3 2+4+5+6+7+8+9 41
16 7+9 1+2+3+4+5+6+8 29
17 8+9 1+2+3+4+5+6+7 28
36 1+2+3 4+5+6+7+8 9 39 6
7 1+2+4 3+5+6+7+8+9 38
23 6+8+9 1+2+3+4+5+7 22
24 7+8+9 1+2+3+4+5+6 21
4 10 1+2+3+4 5+6+7+8+9 35 5
111+2+3+5 4+6+7+8+9 34
295+7+8+9 1+2+3+4+6 16
30 6+7+8+9 1+2+3+4+5 15

You will find familiarity with these blocks extremely useful in solving kakuro puzzles. You can jot down in any block which digits are allowed. Then when two clues cross, you can see if cell at the intersection is now uniquely defined. Consider two clues in which a sum of 11 in four crosses a sum of 29 in four. The unique sets are {1,2,3,5} and {9,8,7,5}, so the common cell can only be 5.

The second most useful tip for solving kakuro puzzles is to look for the maximum and minimum value for the intersecting cell where two clues cross. Even when neither clue is a kakuro block, you can sometimes find a limit to the value of the cell where they cross. For example, when 8 in three cells crosses 28 in four cells, the smallest value for the crossing cell is 4 and the largest value is 5. This is because the smallest digit component of the 28 in four is 4 (9+8+7+4}, while the largest digit component of 8 in three is 5 {1+2+5}.

Lastly, there are also a few intersections of sums which define a unique value for the common cell in which neither of the sums are formed from a unique set of digits. For example when 5 in two cells crosses 21 in three cells, the common digit must be 4. It is easy to see why by looking at the sets of possible digits that make up each sum, then by studying which two sets have a single digit in common. For example you can make 5 in two by using {1,4} or {2,3}, and you can make 21 in three by using {9,8,4} or {9,7,5}; but the only digit these sets have in common is 4. Here is a complete table of all the unique intersections that are possible in kakuro.

### Generating kakuro puzzles Although I enjoy solving kakuro puzzles, I also found it very interesting to find out how to generate them. Like my previous work in generating su doku puzzles, I found that generating good puzzles that are interesting to solve and have a unique solution takes some care. I have been dismayed to find a number of sources of kakuro puzzles on the web in which the puzzles have multiple solutions. You may think you are stuck whereas in fact there is no path to a single answer.

I can now generate kakuro puzzles over a range of sizes and levels of difficulty. Look out for Killer Kakuro a book of my puzzles from Orion Press. If you want to purchase puzzles for publication,

### Sudokuro - a sudoku-kakuro hybrid puzzle As part of my work on sudoku puzzle books I designed a number of sudoku puzzle variants. When I looked at kakuro, it occurred to me that I could make a kind of sudoku-kakuro hybrid that borrowed the look of kakuro and some rules from sudoku.

Sudokuro is played on an 11x11 grid of cells in which each row and each column contains 9 cells and 2 blanks. The use of horizontal and vertical clues to the sum of the blocks is just the same as kakuro, but there is now the added restriction that each row and each column of the whole puzzle can only contain one instance of each digit (or conversely, each digit must appear once in each row and column).

You can find some sudokuro puzzles in Killer Kakuro. Here are some free kakuro puzzles for you to enjoy: Booklet of small kakuro puzzles. Eight small puzzles, copyright free for school use. [answers] Monster 30x20 Kakuro puzzle - fills a whole A4 page! [solution] Sample Sudokuro Hybrid Puzzle. Remember that here, a digit cannot be repeated in a row or column. [solution]     