Han’en Puzzles
Last year, I discovered Sudoku and Kakuro puzzles. I began wondering what a blend of the two might be like. A few months ago, I devised a new type of puzzle that fit the bill. I call it Han’en.
Here is an example of a very easy Han’en puzzle and its solution:
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Every Han’en puzzle is played on a 6×6 grid. The puzzle is solved when:
- Every cell in the grid contains a digit between 1 and 9.
- No digit appears more than once in any row or column.
- Every digit appears in exactly four cells of the grid.
- The digits in each row (or column) add up to the number shown to its right (or below).
Let’s compare Han’en with Sudoku.
- A solved Sudoku is a 9×9 array of digits. A solved Han’en is a 6×6 array of digits.
- In a solved Sudoku, each row and column contains a single appearance of every digit between 1 and 9. In a solved Han’en, each row and column contains a single appearance of some set of six of the digits between 1 and 9.
- Each digit from 1 to 9 appears exactly nine times in a Sudoku, but only four times in a Han’en.
The sum of all the digits in a solved Sudoku is 405 (9×1+9×2+…+9×9). In a solved Han’en, the overall sum is 180 (4×1+4×2+…+4×9). 180 is the number of degrees in a semicircular arc, hence the name Han’en, which is Japanese for semicircle.
Now let’s compare Han’en with Kakuro.
- The clues of a Kakuro puzzle are sums of partial rows and columns. The clues of a Han’en puzzle are sums of complete rows and columns.
- Unlike Kakuro, but like Sudoku, a Han’en puzzle contains a sprinkling of digits from the solution as additional clues.
There is no reason for a Sudoku puzzle to include row and column sums. They are always the same number, 45. The sums in a Kakuro or Han’en can vary, and are needed to solve the puzzle.
It is possible for all the row and column sums of a Han’en puzzle to be the same. I call Han’ens of this type perfect. For the overall total to remain 180, every row and column sum in a Perfect Han’en must be 30. Perfect Han’ens are the most Sudoku-like of the Han’ens.
Here is an example of a very easy Perfect Han’en puzzle, and a link to its solution:
| 1 | 8 | 30 | ||||
| 2 | 3 | 5 | 9 | 30 | ||
| 6 | 5 | 30 | ||||
| 7 | 8 | 1 | 3 | 30 | ||
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6 | 8 | 2 | 4 | |
30 |
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1 | 9 | 30 | |||
| 30 | 30 | 30 | 30 | 30 | 30 | 180 |
In future posts, I will present other Han’en puzzles, including some that I have only been able to solve using a non-obvious special technique. I will leave it to mathematically-inclined readers to rediscover the technique, and I’ll provide a place for them to explain it when they have.