Modeless in Cyberspace

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:

Every Han’en puzzle is played on a 6×6 grid. The puzzle is solved when:

1. Every cell in the grid contains a digit between 1 and 9.
2. No digit appears more than once in any row or column.
3. Every digit appears in exactly four cells of the grid.
4. 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.

1. A solved Sudoku is a 9×9 array of digits. A solved Han’en is a 6×6 array of digits.
2. 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.
3. 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.

1. 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.
2. 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:

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.

This entry was posted on Sunday, June 11th, 2006 at 12:44:24 and is filed under Han'en Puzzles. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.