Unsolvable Rubik's Cube: 3 Assembly Mistakes and Fixes

First: is it the cube, or a mistake in the steps?

Most of the time, a cube that "won't solve" comes from a mistake in an earlier step, not from the cube itself. Before you think about taking it apart, ask yourself:

• Has a piece ever popped out (flown off while you were turning fast), or has someone taken the cube apart and put it back together?
• Is any corner sticking out or sitting crooked compared with the other pieces? Ruwix notes that a normal cube that won't solve is often just a corner that has been twisted (Ruwix).
• Does the cube have stickers that were ever peeled off and stuck back on?

If the answer to all three is "no", solve it again from the start, slowly, following how to solve a 3x3 Rubik's Cube for beginners. If any answer is "yes", read on.

Why can't a wrongly assembled cube be solved?

When you only turn the faces, the cube always follows three rules. No move sequence can break them:

1. Corner rule: you can't twist just one corner in place. Twisted corners always "cancel each other out".
2. Edge rule: you can't flip just one edge. The number of flipped edges is always even.
3. Swap rule: you can't swap exactly two pieces and leave every other piece where it is.

Wikipedia explains it by counting: if you include every way of taking the cube apart and putting it back, there are 12 times as many ways to arrange the pieces as there are states you can reach by turning, and only one in twelve of them can be solved (Wikipedia). The 12 comes from the three rules above: the corner rule divides by 3, the edge rule by 2 and the swap rule by 2. Ruwix works it out the same way: if you put the pieces back at random, only about 8% (1 in 12) can still be solved (Ruwix).

We checked all three rules on the simulator with 200 random scrambles: none of them broke a rule.

One important thing: the simulator can't draw a wrongly assembled cube, because every picture on this page is made from real turns. So each section below shows you the closest valid case, so you can compare it with your cube and notice when yours looks different.

Mistake 1: one twisted corner

Where you'll notice it

This mistake shows up in step 5 of the guide, the step where you make the yellow face. In that step, if you count the corners with a yellow sticker on top, you can only ever see 0, 1, 2 or 4 of them. If you see 3 yellow corners on top and exactly one with its yellow sticker facing the side, the cube has a twisted corner: by the corner rule, that state can't happen on a correctly assembled cube.

An example of a valid state: one yellow corner on top, and the other three all twisted the same way. The Sune algorithm solves it:

R U R' U R U2 R'

SpeedCubeShop also lists "one twisted corner" as a case that can't happen on a 3x3 or on any NxN cube, and says it comes from wrong assembly or from a corner accidentally getting twisted while turning (SpeedCubeShop).

Two twisted corners can still be solved

Unlike one corner, two corners twisted in opposite directions is a valid state. Hold the cube with yellow on top and the corner you want to twist at front right, then do:

R' D' R D

Do this set twice and the front-right corner twists one way; do it four times and it twists the other way. The lower layers get mixed up for a while, don't worry. Turn U to bring the second corner to front right, do the set four times, then turn U back. The lower layers go back to how they were by themselves.

In the picture above: on the left, exactly two corners on the top face are twisted; on the right is the solved cube. On the simulator, doing R' D' R D six times brings the cube back to where it started, so the number of times here can only be 2 or 4. If your cube has only one twisted corner, this method will never finish: you'll always have one corner left over.

How to fix it

According to SpeedCubeShop, the quickest way is to grip the crooked corner with your thumb, index finger and middle finger and twist it clockwise or anticlockwise until it faces the right way (SpeedCubeShop). A few tips:

• Solve the cube as far as you can first, so that only that one corner is wrong. Then twist just that corner.
• A loose speed cube is easier to twist. A stiff cube may need the screw under a centre loosened first, or an edge next to it taken out.
• Twist slowly. Twisting too hard can snap the corner's foot.

Mistake 2: one flipped edge

Where you'll notice it

This mistake shows up in step 4 of the guide, the yellow cross step. Looking only at the four edges on the top face, the number with yellow facing up is always 0, 2 or 4. If you see 1 or 3, the cube has a flipped edge.

All three valid cases use the same algorithm:

F R U R' U' F'

Horizontal line (2 yellow edges on top), do it once:

L shape in the back-left corner (2 yellow edges on top), do it twice:

Just the centre dot (0 yellow edges on top): do it once, turn U2, then do it twice:

Each time you do the algorithm, the number of yellow edges facing up goes up or down by 2, never by 1. So if you start with 3, you can only go 1, 3, 1 forever and never reach 4. Every case is covered in detail in How to solve the last layer of the Rubik's Cube.

How to fix it

You have to pop that edge out and put it back the other way round:

1. Solve the cube until only the flipped edge is left, then turn it so the edge is in the top layer.
2. Turn the top face about 45 degrees (half a U move).
3. Use your fingers or something flat and stiff but not sharp (like a plastic card) to lever up the edge that is showing in the gap.
4. Flip the edge over, press it back into its spot and turn the top face straight again.

SpeedCubeShop describes exactly this way of taking the piece out: bring it to the top layer, turn the top layer 45 degrees and pull it out, sometimes with a bit of force (SpeedCubeShop). With a magnetic cube or a very tight cube, loosening the screws first makes it less likely to break.

Mistake 3: two pieces swapped

Where you'll notice it

With the beginner method, this mistake usually shows up in step 7 of the guide. Every other piece is already right, and only two edges on the top layer have swapped places (next to each other or opposite). The U-perm in step 7 always moves exactly three edges around in a cycle, so no matter how many times you do it, it can't swap just two.

R U' R U R U R U' R' U' R2

The picture above shows a valid case: three edges out of place in a cycle. Compare it with your cube: if only two pieces are wrong and every other piece is right, the cube has been assembled wrong.

On a correctly assembled cube, swaps always come in pairs. For example, the T-perm swaps two corners and two edges at the same time:

R U R' U' R' F R2 U' R' U' R U R' F'

Swapping two corners together with two edges is valid. Swapping only two edges, or only two corners, is not. You can read more about the T-perm in The T-perm algorithm.

How to fix it

SpeedCubeShop suggests solving the cube first, finding the two pieces in the wrong places, bringing each one to the top layer, popping them out as in mistake 2, swapping them and putting them back (SpeedCubeShop). If the two wrong pieces are corners, it works the same way, but corners are harder to pop out; many people choose to take the whole cube apart and put it back together from scratch.

Note: on a 4x4 or bigger cube, "two edges swapped" on the last layer is normal. It's called parity and has its own algorithm. See How to solve a 4x4 Rubik's Cube for beginners. This post is only about the 3x3.

Sticker mistakes

Cubes with stickers have one more kind of mistake: stickers peeled off and stuck back in the wrong place. Ruwix and SpeedCubeShop list the signs that tell you for sure the stickers are wrong (Ruwix, SpeedCubeShop):

• Two centres have the same colour.
• One piece has two stickers of the same colour.
• One piece has the colours of two opposite faces, for example a white and yellow edge.
• One colour has a number of stickers other than 9.

If you see any of these signs, no amount of taking the cube apart will fix it; you need to re-stick the stickers to match a solved cube, or buy a new sticker set. Ruwix works out that if you stick the 54 stickers on at random, the cube is almost certainly unsolvable.

The surest fix: take it apart and rebuild it solved

When you don't know where the cube is wrong, or there are several mistakes at once, Ruwix suggests taking the cube apart and putting it back together with every piece in its solved position (Ruwix). If you rebuild it solved, all three rules are automatically right.

• Never put the pieces back at random: only 1 time in 12 will you be lucky enough for it to be solvable.
• Put the bottom layer in first, each piece matching the colours of the two centres next to it, then the middle layer, and the top layer last.
• When you're done, do a few turns and then undo them to make sure the cube still goes back to solved.

Summary table

Each algorithm comes with its picture. Hold the cube with yellow on top for all of them.

• Yellow cross (step 4): F R U R' U' F'

• Sune (step 5): R U R' U R U2 R'

• Twist two corners: R' D' R D twice, U, four times, U'

• U-perm (step 7): R U' R U R U R U' R' U' R2

• T-perm: R U R' U' R' F R2 U' R' U' R U R' F'

Stuck? Let EasyRubic check

With the EasyRubic app, you can scan your real cube with the camera and see the solution step by step on a 3D cube. Get EasyRubic and practise on your own cube.

Sources

• Unsolvable Rubik's Cube Scrambles – Ruwix
• Unsolvable Rubik's Cubes? Impossible Cubes – SpeedCubeShop (Ari)
• Rubik's Cube – Wikipedia