How to solve
a 3D sudoku cube?

The rest of this article goes through each reflex, with figures measured on our cubes and two real examples.

Sudoku-Land's Classic Cube is built like a Rubik's Cube where each small cube on the surface carries a single digit. Depending on where it sits, a small cube shows on one, two or three faces. Each face is still a normal 9×9 sudoku, but a border cell must follow the rules of all its faces:

In all, 386 cells instead of 6 × 81 = 486. A middle cell “sees” 20 other cells, as in an ordinary grid: those in its row, its column and its box, which cannot hold the same digit. An edge cell sees 32, a corner 36. More neighbours means more digits to rule it out: border cells are the most constrained on the cube. And the 92 border cells are what link the faces together: every piece of information passes from one face to another through them.

Take one face of a cube, with only the digits it shows at the start, and forget the rest of the cube. Here is what you get out of it:

From Medium level on, an isolated face leads nowhere: of the 600 faces of our Medium cubes, none has a unique solution on its own, and you only find 2 cells out of 61 before getting stuck. Its neighbours bring it the digits it is missing. So you do not fill in six grids one after the other: you work your way around the cube.

An edge cell belongs to two faces: it must respect its row, its column and its box on both of them. So a digit already placed on the neighbouring face can rule it out, even though nothing rules it out on the face you are looking at. This is the deduction that belongs to the cube, and it comes up all the time.

The example

Here are two neighbouring faces of one of our Medium cubes, at the very start of the game, unfolded side by side: face 1 and, to its right, face 2. Their edge is tinted: column 9 of face 1 and column 1 of face 2 are the same nine cells, seen from both sides. The 9 and the 5 on it actually show on both faces.

A real Medium cube, at the start. On face 1 alone, the 4 has two places in the outlined box.
Face 2 rules one out: its row 5 runs through the edge cell.

On face 1, let's look for the 4 in the outlined box, middle right. The 4 in row 4 and the 4 in column 7 rule out almost the whole box: two places are left, R5C9 (row 5, column 9) and R6C8. On this face alone, there is no way to choose.

But R5C9 is on the edge. On face 2, it is cell R5C1, and row 5 of face 2 already has a 4. So the 4 cannot go in R5C9: it goes in R6C8, on face 1.

Notice where the answer lands: in the middle of the face, not on the edge. That is the most common case. On our cubes, when a cell can only be found by looking at the neighbouring face, it is in the middle of its face 6 times out of 10: the neighbouring face rules out the border cells, and a middle cell is the one left. At the start of a Medium cube, a player who looks at one face at a time finds 8 to 9 playable cells on average; by also looking at the neighbouring faces, they find 14 more.

1. Start with the fullest face. Lay it flat and play it like an ordinary sudoku, box by box. At Medium level, 87% of the cells are found this way, by looking at a single face.
2. When the face gets stuck, check its borders. For each digit torn between two places, one of which is on an edge, look at the neighbouring face. The arrows on the edges of the flat face bring it up without going back to the whole cube.
3. Switch faces often. A digit placed on an edge immediately shows on the neighbouring face, and can unlock it. At Medium level, it happens 4 times per cube on average (up to 7) that no face, taken on its own, shows anything more: that is your cue to turn the cube.
4. Keep an eye on corners and edges. They are the most constrained cells: at Medium level, a corner is found on average a quarter of the way through the game, an edge cell a little later, a middle cell halfway. Borders filled in early then feed the neighbouring faces.
5. Finish the faces one at a time, in no fixed order. A face filled in without errors turns gold and locks its cells, including the ones it shares with its neighbours: those digits are settled for good. But do not keep at a face that resists: the next step will often come from elsewhere.

At Hard level, looking at both faces is no longer enough: our 100 Hard cubes all need at least one technique beyond singles. Solved by the game's hint engine, they call on locked candidates in 99 cubes out of 100, a hidden pair in 43, and two techniques chained together in 37.

The good news: these techniques work across the edges. A box against an edge shares its three border cells with a box on the neighbouring face. If, in one box, a digit can only go in those cells, the neighbouring box necessarily gets it in the same place: you can remove it from its other cells. In nearly one Hard cube in three, the hint engine uses such a lock from one face to another at least once.

The example

This Hard cube is mid-game. On face 2, in the top right box, the 8 in column 7 and the 8 in column 8 rule out those two columns: the 8 of the box must be in column 9, in other words on the edge, in R1C9 or R3C9 (the small blue 8s).

A real Hard cube, mid-game.
The lock is read on face 2, its effect on face 3.

Across the edge, these two cells are R1C1 and R3C1 of face 3, in its top left box. So the 8 of that box is also in one of them, and nowhere else in the box: you can remove it from cell R2C2.

Now look at row 2 of face 3. It has four empty cells left: R2C2, where we have just removed the 8, R2C3 and R2C9, ruled out by the 8s in their columns, and R2C8. The 8 goes in R2C8, at the far end of the face, thanks to a lock read on the face next door.

The same reasoning works with a line: if the 8 of a box can only go on the edge, it also disappears from the rest of the edge, on both its faces.

The Ultimate Cube keeps the shared edges, but each face has its variant: Sudoku X, Hyper Sudoku, even-odd, jigsaw, greater than and Killer. A border cell then follows the rules of both its faces. On the border between the even-odd face and the Killer face, for instance, a digit must both have the right parity (shaded cell = even digit) and fit the sum of its cage.

Two consequences for the method:

• Start with the well-stocked faces. The Killer and greater than faces start with few digits, or none at all: their borders fill in as you make progress on their neighbours, and they are what open them up.
• A rule is read on its own face. A cell's shading, its signs and its cage only show on the face of their variant. On the neighbouring face, the same cell looks ordinary: when in doubt, turn the cube to read its whole rule.

On the Variant Cube, all six variants are there, but the faces are separate: each one can be solved on its own, like a puzzle from the variant pages. So there is no method specific to the cube: reread each face's rule, shown above the number pad, and apply the method of its variant. For the Killer face, our article How to Solve a Killer Sudoku explains the rule of 45; the Variants page introduces the others.

The game's Hint button follows the same approach as this article, in three steps: where to look, then which technique to use, then the answer, explained. When the reason comes from a neighbouring face, it says so, and a button turns the cube to show it to you. Reading a hint is free: only “Apply” uses up one of the cube's six hints.

Where do you start a sudoku cube?

With the face that has the most digits, played like an ordinary sudoku. As soon as it gets stuck, check the cells on its borders against the neighbouring face: at Medium level, one cell in eight can only be found that way.

Can you solve the faces of the cube one by one?

Not on a cube with shared edges (Classic Cube, Ultimate Cube). At Medium level, no face of our cubes has a unique solution on its own: its neighbours bring it the digits it is missing. On the Variant Cube, on the other hand, the faces are separate and can be solved one by one.

Which techniques do you need for the Hard level?

Singles are no longer enough: all our Hard cubes need at least one more technique, mostly locked candidates, sometimes a hidden pair or two techniques chained together. They also work across the edges: a box against an edge shares three cells with a box on the neighbouring face.

How many cells does a sudoku cube have?

386 on Sudoku-Land's Classic Cube: 294 in the middle of the faces, 84 on the edges (visible on two faces) and 8 corners (visible on three faces), instead of 6 × 81 = 486 if the faces were separate.

The three cubes are free to play on the 3D Cube page, with no sign-up, on desktop and mobile alike. For this method, start with the Classic Cube at Easy level: each face is well stocked there, and moving back and forth between the faces comes gradually. There are 100 different cubes per level, and the game always gives you a cube you have never started first.

To brush up on the techniques of classic sudoku, the basis for everything else, read the complete method.