In base-4 sudoku, such a Determinant Sudoku is a grid of 12x12 numbers, not too much larger than a base-10 ordinary sudoku. But I wasn't sure if any valid base-4 Determinant Sudoku actually existed, so I wrote a program to help me check.

I did manage to construct a valid base-4 Determinant Sudoku, but I'm not sure how many others exist.

Here is the source code to enumerate all base-4 sudoku and print their determinants, and I've copied the comment header below which includes more explanation and the base-4 determinant sudoku I found:

// This program enumerates all the base-4 sudoku and calculates

// their determinants.

//

// The idea is this: start with base-4 sudoku, so that you get

// 2x2 squares which have the numbers 0,1,2,3 with none repeated.

// Create a 2x2 grid of those 2x2 squares, so that each row and column

// has the numbers 0,1,2,3. (Ordinary sudoku, but 4x4 instead of 9x9).

//

// Here's an example of a valid base-4 sudoku:

// 01 23

// 23 01

//

// 10 32

// 32 10

//

// There appear to be 288 distint base-4 sudoku, without removing isomorphisms.

//

// My idea was to take the determinant of each of those 2x2 matrices and create

// a "determinant sudoku".

//

// In matrix math, the determinant of

// a b

// c d

// is a*d - b*c.

//

// There are six possible determinants of the 2x2 blocks.

//

// To construct a base-4 determinant-sudoku, make a 3x3 grid of 4x4 base-4

// sudoku blocks. There will be six rows and six columns of 2x2 squares.

//

// How many base-4 determinant-sudoku blocks exist?

// This program almost answers that question, but not quite.

//

// Of the 288 base-4 sudoku, 168 have the same determinant, leaving 120

// base-4 sudoku that could contribute to a base-4 determinant-sudoku.

//

// I constructed one base-4 determinant-sudoku by hand:

//

// 01 23 | 01 32 | 02 31

// 23 01 | 32 01 | 31 02

// | |

// 10 32 | 10 23 | 20 13

// 32 10 | 23 10 | 13 20

//

// ----------------------

//

// 01 32 | 02 31 | 01 23

// 32 01 | 31 02 | 23 01

// | |

// 10 23 | 20 13 | 10 32

// 23 10 | 13 20 | 32 10

//

// ----------------------

//

// 02 31 | 01 23 | 01 32

// 31 02 | 23 01 | 32 01

// | |

// 20 13 | 10 32 | 10 23

// 13 20 | 32 10 | 23 10

//

//

// Determinants:

//

// -2 2 -3 3 -6 6

// 2 -2 3 -3 6 -6

//

// -3 3 -6 6 -2 2

// 3 -3 6 -6 2 -2

//

// -6 6 -2 2 -3 3

// 6 -6 2 -2 3 -3

//

// Each 4x4 block is a valid base-4 sudoku. And the 6x6 grid of 2x2 blocks

// has no repeated determinants in any row or column.

//

// (This base-4 determinant sudoku isn't very nice, since it repeats the

// same 3 base-4 sudoku 3 times each.)

//

// So the question remains open: how many base-4 determinant sudoku are there?

## No comments:

Post a Comment