Problem of the Week #11

Spring 2006: Special All-Sudoku Semester

A regular Sudoku board is a 9x9 grid with rows, columns, and 3x3 subsquares as regions, and 9 numbers are needed to populate the cells. This week we kick it up a dimension! The smallest true 3-dimensional generalization of Sudoku is an 8x8x8 cube, whose regions are the 8x8x1 "slices" of the cube and certain 4x4x4 subcubes. Notice that 8^2 = 4^3 = 64, so 64 symbols would be needed to populate the cells! Since that is a bit prohibitive, we'll do something a little different for this week's puzzle: a 4x4x4 cube with slightly modified Sudoku rules.

In this puzzle the subregions are again slices and subcubes, but notice that the slices are size 4x4x1 = 16, while the subcubes are size 2x2x2 = 8. Therefore we will use 8 symbols to populate the cells, filling in the puzzle so that each subcube uses each symbol once, while each slice uses each symbol twice. Here's an example to get you started on thinking about the regions: Consider the second slice back from the front left face of the cube (the slice with "8," "2" and "4" showing around its edge on the picture of the cube). The 16 cells in this slice correspond to the cells in the four "third rows" of the given 4x4x4 grids. Try considering how all types of slices and subcubes look in the 4x4x4 grids before you begin trying to solve the puzzle!

Rules: The four 4x4 grids represent the horizontal slices of the cube, from top to bottom. Fill in with 1-8 so that each of the marked 2x2x2 subcubes contains each number once, and each 4x4x1 slice of the cube contains each number TWICE.

Questions to ponder: Why is an 8x8x8 cube the smallest cube that allows a "true" 3-d generalization of Sudoku? (Here "true" means that everything jumps a dimension: rows and columns are slices of a square, and are generalized to slices of a cube, while subsquares are generalized to subcubes.) What is the next smallest size of cube that would work?

Another puzzle to try: In 2005 the Daily Telegraph published a 9x9x9 Sudoku cube that they called the "Dion cube". This puzzle is really 27 interlocking Sudoku boards, and not a "true" 3-d generalization, but that doesn't mean it isn't fun to play. You can find it at www.sudoku.org.uk/PDF/Dion_Cube.pdf. Once you find (or look up) the solution, you'll notice that it is extremely regular. Does this mean that there aren't very many "Dion cube" puzzles? Or was the puzzle creator just including a nice pattern?

Solutions are due by noon on Tuesday, April 4, 2006.
Return solutions ON PAPER to Laura Taalman, Burruss 127, MSC 7803.
Include your name and email address with your solution.