## Cube Sudoku

 MathRec Home Sudoku of the Day Sudoku Home Sudoku Archive Cube Sudoku Archive Last Month's Topic Introductory Topic Older Topicsand Links Educational Resources Steve's Personal Page Sudoku

In a cube sudoku puzzle, the nine 3×3 boxes are replaced by six 4×4 boxes on the surface of the cube. The rows and columns of a standard sudoku grid are replaced by strips that run all the way around the cube. Each of the strips crosses four faces, with four cells on each face. In this way, each face and each strip has sixteen cells. The entire puzzle has 96 cells.

Because the cube is fundamentally three-dimensional, it's a bit difficult to represent on a flat web page. I've settled on the format you see below. If you consider the puzzle carefully, you should see that the faces form a flattened cube.

There is no reason that a cube sudoku cannot be solved on the surface of an actual cube, turning the cube over in your hands as you work the puzzle. I've created actual cubes by trimming the printed puzzle, gluing it onto a poster board, then cutting, folding and gluing the poster board. The resulting puzzle is sturdy enough to write on, but difficult to erase.

The 2009 World Sudoku Championships recently featured a cube sudoku puzzle as part of the finals competition.

Cube Sudoku is just one of a series of variants that you can make if you allow the "rows" and "columns" to bend. Other variants are currently featured on the home page and new puzzles are posted daily on Sudoku of the Day.

#### February 9, 2009

I'm trying to be a little more regular about posting new sets of cube sudoku. My target is to get out a new set every two or three weeks. If that's not enough, let me know!

Easy and Moderate puzzles require only elimination. Subtle, Tough and Challenging puzzles require block interactions (see discussion below). The Strenuous puzzles are more difficult than the Challenging puzzles, but do not require any techniques beyond block interactions.

Place each of the numbers 0–9 and each of the letters A–F exactly once in each strip and each 4×4 face of the flattened cube.

Difficulty: Easy   Larger Puzzle and Solution
Other Difficulties:
Moderate and Solution
Subtle and Solution
Tough and Solution
Challenging and Solution
Strenuous and Solution
Extra Strenuous and Solution

There are several reasons that I enjoy cube sudoku:

• It is a welcome change from the standard 9×9 grid of 3×3 boxes.
• It has 4×4 boxes with only 96 cells, instead of the 256 cells in a 16×16 grid.
• The block interactions are more complex because the strips wrap around the puzzle.

The last comment deserves a little explanation. Until I write a full page about it, here is a brief summary:

There are three sets of strips. Referring to the flattened cube above, I call them rows (red), columns (green), and rings (blue). Each set has four parallel strips. I say that they are parallel, because they never intersect. If, however, you consider two rings from different sets, say rows and rings, then you see that they intersect in two cells located on opposite faces. In the case of rows and rings, one intersection is on the left face and the other intersection is on the right face. This double intersection allows a more complicated type of block interaction.

In standard sudoku, you can have a block interaction when all of the possible locations for a certain value in a 3×3 box are in a single row. When this happens, you can conclude that the value cannot occur in any other cell of that row that is outside of that 3×3 box, because it would lead to an immediate contradiction. The same interaction can occur with columns instead of rows, and it can work in reverse (when all of the possible locations for a certain value in a row are in a single 3×3 box).

In cube sudoku, you get the same type of block interaction when all of the possible locations for a certain value on face are located in the same strip. This is a two-block interaction involving one face and one strip. But you can also get the same type of interaction when all of the possible locations for a certain value on a face are located in two intersecting strips. In this case, you can eliminate that value from the other intersection of those two strips. This is a three-block interaction involving one face and two strips. You can also get two-block interactions involving two strips and three-block interactions involving three strips.

The first cube sudoku (below) was created by hand. As a result, it was solvable using reasonable logical deductions. My second cube sudoku (not Cube Sudoku #2, below) was created by a generator that I wrote. That turned out to be a brutal puzzle. I needed ten sheets to solve it by hand, following different forks to find the solution and eliminate the alternative choices. After that, I fixed the generator to create puzzles that could be solved with elementary techniques. Cube Sudoku #4 and Cube Sudoku #6 require block interactions. The Moderate puzzles are solvable using elimination alone, but may be much easier if you also use block interactions and/or pairs.

#### Previous Cube Sudoku Puzzles

Cube Sudoku #7 (Easy) and solution
Cube Sudoku #6 (Tough) and solution
Cube Sudoku #5 (Moderate) and solution
Cube Sudoku #4 (Tough) and solution
Cube Sudoku #3 (Tough, but can be performed with elimination alone) and solution
Cube Sudoku #2 (Challenging, but can be performed with elimination alone) and solution
Cube Sudoku #1 (Moderate) and solution

#### Other Sudoku Puzzles and Links

Code-Doku Puzzles -- Another sudoku variant with a hidden message in the puzzle.
Code-Doku Archive -- Previous codedoku puzzles from this site.