View source for Gurth's Symmetrical Placement

'''Gurth's Symmetrical Placement''' is a [[technique]] for solving [[Sudoku]] puzzles with 180-degree rotational symmetry in the [[given]]s, i.e., the cell '''r[i]c[j]''' contains a given if and only if the cell '''r[10 - i]c[10 - j]''' also contains a [[given]].

The technique is applied to a rotational symmetric puzzle as follows. Suppose for every digit '''n''', we can find a digit '''k''', such that for every cell in this puzzle that contains the given '''n''', its opposite cell (i.e., the cell that is rotated about '''r5c5''') contains the given '''k'''. Then all the other cells in this puzzle will also contain this property. Furthermore, if the '''r5c5''' cell is empty, then '''r5c5''' can be assigned the digit '''n''' whose opposite digit is also '''n''', or the missing digit if the puzzle contains givens for only eight of the nine digits.

Stated another way, let the nine digits of a rotational symmetric puzzle be '''a1''', '''a2''', '''b1''', '''b2''', '''c1''', '''c2''', '''d1''', '''d2''' and '''e'''. Suppose also that for all '''i''' and '''j''' in {1, ..., 9}, whenever the cells '''r[i]c[j]''' and '''r[10 - i]c[10 - j]''' are non-empty, we have both cells containing {'''a1''', '''a2'''}, {'''b1''', '''b2'''}, {'''c1''', '''c2'''}, {'''d1''', '''d2'''} or {'''e''', '''e'''}. Then the empty cells can be filled up so that the above property is satisfied. Also, if '''r5c5''' is empty, then we can assign '''e''' to it.

The rationale for '''Gurth's Symmetrical Placement''' is that if we can apply a technique '''T''' to a group of cells '''S''' to assign a digit '''X''' to some particular cell '''A''', then we can also apply the same technique '''T''' to the cells that are opposite to '''S''' and assign the digit that is the partner of '''X''' to the cell that is opposite to '''A'''. The puzzle is required to contain a [[unique]] solution for '''Gurth's Symmetrical Placement''' to be valid. 

== Example ==

[[Image:Gurth-example.png]]

The above puzzle is difficult without using '''Gurth's Symmetrical Placement'''. However, once '''Gurth's Symmetrical Placement''' is applied, the remainder of the puzzle can be solved by [[single]]s alone. More specifically, the digit pairs are {'''1''', '''2'''}, {'''3''', '''6'''}, {'''4''', '''7'''}, {'''5''', '''8'''}, and the missing digit '''9'''. (For example, '''r3c4''' and '''r7c6''' are opposites and contains '''1''' and '''2''', and '''r6c9''' and '''r4c1''' are also opposites and also contains {'''1''', '''2'''}.) So we can place '''9''' in '''r5c5''', and the rest is very easy.

[[Category:Solving Techniques]]
[[Category:Uniqueness]]