Mirrored from **Sudopedia**, the Free Sudoku Reference Guide

**Gurth's Symmetrical Placement** is a technique for solving Sudoku puzzles with 180-degree rotational symmetry in the givens, 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.

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 singles 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.