Problem of the Week #9

Spring 2006: Special All-Sudoku Semester

One of the simplest and most popular variants of Sudoku is called "Sudoku X", or sometimes "Sudoku Extreme." Entire puzzle magazines are devoted to this variation, in which the two main diagonals of the Sudoku board are also Sudoku regions (i.e. contain all of the integers from 1 to 9). The "Snowflake" variation below has six diagonal regions in which no number can be repeated. Notice that some of the marked diagonals contain less than nine entries and thus are not full Sudoku regions; this is in fact the key to what makes this puzzle interesting.

Rules: Fill in the grid with the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 so that each row, column, block, and marked diagonal has no repeated entries.

Questions to ponder: How many diagonal regions can be incorporated into a Sudoku board? For example, does a Sudoku board exist that contains each of the 17 possible increasing diagonals? If so, how many such boards exist? How about boards that contain each of the 34 possible increasing and decreasing diagonals? What if the diagonals are "toroidal" (i.e. "wrap around" the board, so each diagonal is effectively of size 9)? How does the number of diagonal regions affect the number of initial conditions needed to ensure a unique solution?

Solutions are due by noon on Tuesday, March 21, 2006.
Return solutions ON PAPER to Laura Taalman, Burruss 127, MSC 7803.
Include your name and email address with your solution.