## Sudoku viewed as 3D with (x,y,z) notation

Advanced methods and approaches for solving Sudoku puzzles

### Sudoku viewed as 3D with (x,y,z) notation

I am a newbie here, and just a recent Sudoku addict, but from a programming approach.
Excuse me if what I say is blindingly obvious and known to everyone from first principles.
We depict the grid as 2D, but it seems very obvious that the object is in fact 3D, with our Z dimension represented in flatland as a small square grid.
My first programming instinct had been to think about row, column and square objects separately, but if I view the grid in X,Y,Z terms then the symmetry becomes much more useful at least from a conceptual viewpoint.
So then Row, Column and Square are all totally interchangeable; any rule which involves Row and Column equally applies to Column and Square , Square and Row.
Starting with A1 as the origin (0,0,0) we can mentally rotate and see the grid in conventional 3D form with I9 at (8,8,8)
We can apply transformations of rotation or shift of origin or interchange of XYZ without impact
Is this XYZ approach a useful way of viewing Sudoku?
laogui32

Posts: 3
Joined: 02 September 2005

Neat!

But why stop at three? Imagine each cell in the 3-D space as being occupied by any of 9 identical coloured objects so a yellow in (2,3,4) precludes a yellow elsewhere in (2,3,4) or in (i,3,4), (2,j,4) or (k,3,4). Alternatively see it as 9 conventional Sudokus with a connection rule between them that has this effect.
Max Beran

Posts: 57
Joined: 17 August 2005

I assume with "square" you mean what we call "block" here,
not to be confused with the "cells" which are also small squares.

But then, the blocks are not interchangeable with rows or columns.
A block intersects 3 rows and 3 columns, a column intersects
3 blocks and one row.

You can consider the value in a cell as the z-coordinate where
a sudoku-piece has to be placed in 3d.
Sudoku-pieces attack each other in rows,columns,piles
and planar 3*3*1 boxes parallell to the xy-plane.
Then a completed sudoku is a configuration of 81
mutually nonattacking sudoku-pieces.

You see, the condition with the 3*3-planes is a bit artificial
and not symmetric. What happens, if we allow sudoku-pieces
to walk along any aligned 3*3*1 box , not only those
in xy-direction ?

Has tso (or anyone else) an example for such a puzzle ?
dukuso

Posts: 479
Joined: 25 June 2005

### 3D

each cell belongs to precisely one row, one column, one 3x3 square
each column intersects 9 rows 3 'blocks' you are right, no symmetry
each row intersects 9 columns 3 'blocks'
each 'block' has 3 intersecting columns and rows, yes so no symmetry
But xyz still looks inherently interesting as programming concept

sorry for my rash assertion of symmetry!
laogui32

Posts: 3
Joined: 02 September 2005

I'm sure you'll find that most Sudoku programmers have considered this option - but in reality it poses more problems than it solves.

stuartn
stuartn

Posts: 211
Joined: 18 June 2005