Gurth's Theorem
Gurth's Theorem — also called symmetrical placement — is a rare but beautiful shortcut. When a puzzle is built with symmetry, that symmetry leaks into its solution, letting you place digits from the shape of the givens alone. It only works on the minority of grids designed this way, but where it does, it can crack a hard puzzle in seconds.
The idea
Suppose the givens are unchanged when you rotate the grid 180° and relabel every digit by the pairing 1↔9, 2↔8, 3↔7, 4↔6, 5↔5. If the puzzle has a single solution, that solution must obey the same symmetry — otherwise its mirror image would be a second, equally valid solution, contradicting uniqueness. Two consequences follow at once. The centre cell maps to itself under the rotation, so it must equal its own pair: only 5 satisfies 5↔5, so the centre is 5. And every other cell is tied to its rotational partner: fix one and its partner is the paired digit. Diagonal-symmetry versions work the same way with a reflection instead of a rotation.
A worked example
Take a puzzle whose clues are perfectly symmetric under a 180° turn with the 1↔9 … 5↔5 relabeling. Straight away the middle cell r5c5 is 5 — no candidates needed. Now if r2c3 resolves to 7, its partner r8c7 must be the pair of 7, which is 3; and r3c6 = 4 forces r7c4 = 6. Each placement hands you its mirror for free, so the grid fills in symmetric pairs.
180° symmetry with 1↔9…5↔5. The centre r5c5 maps to itself, so it must be 5. Each cell pairs with its rotational partner through the centre (dashed): r2c3 = 7 fixes r8c7 = 3, and r3c6 = 4 fixes r7c4 = 6.
Step by step
- Test the symmetry — do the givens survive a 180° rotation with the 1↔9 … 5↔5 relabeling?
- Place the centre — a rotationally symmetric puzzle forces r5c5 = 5.
- Pair the cells — each solved cell fixes its partner as the paired digit.
- Fill in mirrors as ordinary logic resolves each half.
Common mistakes
- Assuming symmetry. The clue pattern and the digit relabeling must both hold — a symmetric shape with mismatched digits does not qualify.
- Wrong pairing. Use the mapping the puzzle is built on; some use a different relabeling, and diagonal symmetry pairs cells by reflection.
- Non-unique grids. The whole argument needs a single solution; never lean on it for a puzzle that might have several.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.