King's Move, Anti-Knight and X-Sudoku: How to Solve the Adjacency Variants
Three variants, one idea. King’s Move, Anti-Knight and X-Sudoku each name a set of cells — on top of the row, the column and the box — where a digit may not repeat a second time.
Because the idea is shared, the technique is too. You are still looking for naked singles and hidden singles; the only thing that changes is which cells count as neighbours. Learn the move once and all three are the same job.
There is one place they stop being alike, and it is worth the wait, so it is at the bottom.
The rules that do not change
Every row, every column and every 3x3 box still holds each of 1 to 9 exactly once, and the puzzle still has one solution reachable without guessing. If that groundwork is new, the rules and the step-by-step guide come first, and naked singles and hidden singles is the technique every deduction below is made of.
King’s Move: the eight cells around a digit
No digit may repeat in any cell a chess king could step to — one square in any direction, diagonals included.
Most of that you already had. Of the eight cells a king reaches, six are normally spoken for: two share the row, two share the column, and five sit in the same box, which comes to more than six because a cell can be caught more than once. What is left is the diagonal neighbours that fall outside the cell’s own box.
On the marked cell, r5c3, exactly two cells are new: r4c4 and r6c4. And that is enough. Ordinary Sudoku narrows r5c3 to a 4 or a 9 and stops. One of the two new cells, r6c4, holds a 4. A king steps diagonally, so the 4 is out and r5c3 is a 9.
How much the rule adds moves around the grid, and it is worth knowing where it adds nothing. A cell in the middle of a box gains not one thing from it: all eight of its neighbours are already in the box with it.
Anti-Knight: the eight cells a knight could jump to
No digit may repeat a chess knight’s move away — two squares one way and one square the other.
A knight’s move always changes both the row and the column, so an Anti-Knight cell never shares a row or a column with the cell it constrains. That makes it almost pure gain: the only overlap left is the box, and it is a small one.
From the marked cell, r2c2, there is no overlap at all. It sits in the middle of its box, and every knight jump from a box’s middle cell lands outside that box. Four of the eight run off the edge of the grid from there — a knight near a corner has fewer places to land — and the four that stay on it are all new.
Ordinary Sudoku leaves r2c2 as a 4 or a 9, exactly as it left r5c3. A 9 sits at r1c4, which is a knight’s move away, so r2c2 is a 4.
Notice how alike the two deductions are. Different geometry, same sentence.
X-Sudoku: the two diagonals
Both main diagonals must also hold each of 1 to 9 exactly once.
The marked cell is r5c5, the centre, and it is the only cell on the board that lies on both diagonals at once. Ordinary Sudoku narrows it to a 1 or a 6. There is a 1 further along the main diagonal at r7c7 and another on the anti-diagonal at r8c2, so either diagonal on its own settles it. r5c5 is a 6.
Where the three stop being alike
King’s Move and Anti-Knight can only ever take candidates away. X-Sudoku can also tell you where a digit must go.
The reason is counting. A hidden single is the argument “this digit has to appear somewhere in this group, and only one cell is left for it”. That argument needs a group that holds all nine digits. A diagonal is nine cells and holds each digit exactly once, so it is such a group, and you can read it for hidden singles exactly as you read a row.
The eight cells around a king are not. Eight cells cannot hold nine digits, and nothing says a particular digit appears among them at all. The same goes for a knight’s eight. So “the 9 has to be in one of these eight, and seven are ruled out” is not a deduction — it is the most inviting mistake in both variants, and it is invalid every single time.
Sudoku Fog’s own solver draws the line in the same place: the king and knight rules are wired to produce eliminations only, and the diagonal rule is wired to produce placements as well.
Try it
A King’s Move board. In Sudoku Fog this is one of the modes you have from the start, with nothing to unlock first — Anti-Knight and X-Sudoku are both unlocked later.
Three ways this goes wrong
Hunting hidden singles in a king’s or knight’s neighbourhood. Covered above, and worth repeating because the reasoning feels identical to the sound version and produces a confident wrong answer. Eliminations only.
Checking all eight cells when six were already covered. For King’s Move the diagonal neighbours in the next box along are the whole of the new information. Anti-Knight is the opposite — nearly every jump is new — which is why it tends to bite harder despite looking more exotic.
Forgetting the diagonals are units when you count. In X-Sudoku a cell on a diagonal has more peers than its neighbours do, and the centre has the most of anyone. If a board feels stuck, the centre is a good place to look.
Where to go next
Kropki and Thermometer are the other two variants worth a page of their own; both work differently from these three, because their clues sit between cells rather than around them.
If you want the version where you cannot see the whole board while you do any of this, fog of war Sudoku is that. Everything published so far is on the Learn index.
Sudoku Fog is on Android — What Sudoku Fog is, Get it on Google Play.