How to Solve a Square-1 - Step-by-Step Course

Before you start

Grab a Square-1 - the best way to learn is to repeat the moves on the real puzzle. You can rotate the animations in this course: drag the puzzle with your finger or mouse. If you know our 3x3 cube course, some ideas will look familiar, but it is not required - everything is explained from scratch.

What the Square-1 is made of

The Square-1 has three layers: the top, the bottom and a thin middle. The top and bottom layers are made of two kinds of pieces:

• 8 corners - wide kite-shaped pieces, 4 on the top and 4 on the bottom of a solved puzzle. A corner has three colours.
• 8 edges - narrow triangles, exactly half as wide as a corner. An edge has two colours.
• The middle - two halves that can only turn relative to each other.

In this course the solved puzzle has a white top and a yellow bottom, a green side at the front, red on the right, orange on the left and blue at the back. If your Square-1 has different colours, nothing changes - read “white” as the top colour and “yellow” as the bottom colour.

Why it changes shape

The top and bottom layers can be turned by any number of “slots” - a slot is the width of an edge, and a corner takes up two slots. A whole layer has 12 slots. The third move is the slice: turning the whole right half of the puzzle by half a turn. When the layers are turned by one or two slots, the slice moves a different number of corners than edges to the other layer - and the puzzle stops being a cube. There are several thousand shapes in total, but you do not need to know them.

The solving plan

1. Shape - you gather 6 corners on the bottom (it looks like a star), and then use one of 5 algorithms to get back to cube shape. This is the only part that needs some thinking - you will get three simple rules.
2. Layer colours - white pieces to the top, yellow ones to the bottom. Two algorithms that swap pieces between the layers.
3. Corners into place - like on the 3x3 cube: you look for “headlights” and swap two corners.
4. Edges into place - cycles of three edges, and sometimes parity as well: a separate algorithm for a situation that cannot happen on a 3x3 cube.

All algorithms in this course keep the cube shape - once the shape is solved, the puzzle will not fall apart again.

1Move notation

How to hold it. Look at the middle layer from the front: the line along which the slice cuts it is not in the centre. Hold the Square-1 so that this line is at the front, left of centre (the narrower part of the middle on the left). Then the slice runs from the front, slightly to the left, to the back, slightly to the right - and it is always the right half that turns. This is how the Square-1 is held in competitions, and this is how all algorithms are written.

(x,y) - turn the top by x slots and the bottom by y slots. A positive number is a clockwise turn when you look at that layer (the bottom - from underneath). In practice: positive for the top is like U on a cube (the front moves to the left), positive for the bottom is like D (the front moves to the right). (1,0) is the top by one slot, (3,0) by 90°, (6,0) by half a turn, (0,-3) the bottom by 90° the other way.

/ - the slice: the right half by half a turn (the direction does not matter, the result is the same). Click a move to see it on the puzzle:

A blocked slice. You can only slice when no corner lies across its line - at the front or the back, on the top or the bottom. On a solved puzzle, after (1,0) the slice goes through, but after (-1,0) it does not: a corner has stopped exactly on the line. Click (-1,0) above and then / - nothing will happen, just like on the real puzzle. That is why the numbers in algorithms have to be exact to the slot: being one slot off usually blocks the next slice.

Try a short sequence: (1,0) / (-1,0) - after it the shape is different, but undoing the same moves in reverse order restores the puzzle.

2Shape: the star

Goal: 6 corners in the bottom layer. The bottom then looks like a six-pointed star, and 2 corners and 8 edges are left on the top. Colours do not matter at this stage.

This is the only stage without a ready-made algorithm, but you do not have to guess. First, two terms:

• Half - the slice divides each layer into two halves of 6 slots. The right half of the top and the right half of the bottom swap during the slice, and they flip over at the same time: what was at the front on the top ends up at the back on the bottom.
• Triple - three corners next to each other, with no edges between them. It takes up exactly half a layer.

Rule 1: triple down

If there is a triple on the top: turn the top so that the triple fills the whole right half. Turn the bottom so that its right half has as many edges as possible (and few corners) and the slice is not blocked. Do / - three corners go down to the bottom, and fewer come back up.

Rule 2: make a triple at the slice line

If there is no triple on the top, make one with a single slice. A triple forms at the slice line: the corners that stay on the top on the left meet the corners that come up from the bottom. Example: on the top at the front left, right next to the slice line, there are two corners, and on the bottom at the back right, also next to the slice line, there is one. After the slice that one comes up to the top at the front right - next to those two. It can also be the other way round: one on the top and two on the bottom.

Before you slice, think about the next slice: choose a join after which the triple can be sent down straight away (rule 1) and there will be more corners on the bottom than now. Usually it is enough that the bottom does not give away more than one corner in the process.

Rule 3: do not undo

A slice right after a slice, without turning the layers, undoes the previous one. If you get lost, do not go back - just start from rule 1 or 2 in whatever state you are in. When no rule fits (it happens rarely), do any slice that does not take corners away from the bottom, and try again.

It usually takes 2 to 5 slices. Over time you will start to see the joins right away - practise in the simulator, where colours do not get in the way, and a blocked slice immediately shows that a layer is turned one slot too far.

3Shape: the cube

Goal: both layers square and a flat middle - the Square-1 looks like a cube (the colours are still scrambled).

There is a star on the bottom, and two corners and eight edges on the top. Count the edges between the corners on the top along the shorter side: 0, 1, 2, 3 or 4. That is the case number. Always set up the bottom so that the slice line passes between the corners of the star (the slice must not be blocked). Set up the top exactly as in the animation - the description is under each one.

Cases 0, 2 and 4 are symmetrical - the corners are equally far from the back slice line. In cases 1 and 3 it is easy to set up the top one slot off; if any slice in the algorithm is blocked, go back and check the setup.

A flat middle

After the algorithm the layers are square, but the middle is sometimes misshapen: the right half is flipped and sticks out. If needed, align the layers (so that the slice can be done) and perform:

From now on every algorithm in the course keeps the cube shape and a flat middle. If something sticks out after an algorithm, a move went wrong - usually by one slot.

4Corners to their layers

Goal: 4 corners with white on the top, 4 with yellow on the bottom (white and yellow facing up or down - there are no other orientations). Edges do not matter yet.

This algorithm swaps two pieces at the front between the layers: the front right corner on the top (together with the edge in front of it) and the front left corner on the bottom (together with the edge in front of it). The other pieces stay in their layers and just shift.

1. Turn the top by 90° so that a yellow corner is at the front right.
2. Turn the bottom by 90° so that a white corner is at the front left.
3. Perform (0,-1) / (0,3) / (0,1).

Repeat until there are no yellow corners on the top - then there are no white ones on the bottom either. At most 4 repetitions are enough, usually 2.

5Edges to their layers

Goal: the whole top white, the whole bottom yellow.

Now we swap single edges: the right edge of the top with the right edge of the bottom. The corners stay in their layers.

1. Turn the top by 90° so that a yellow edge is on the right.
2. Turn the bottom by 90° so that a white edge is on the right.
3. Perform (1,0) / (0,-3) / (0,-3) / (-1,-1) / (1,4) / (0,3) / (-1,0).

Repeat until there are no yellow edges on the top. This is the longest algorithm in this part - memorise it in three chunks: (1,0) / (0,-3) / (0,-3) /, then (-1,-1) /, and finally (1,4) / (0,3) / (-1,0).

After this stage the top is all white and the bottom all yellow. Look from the side: the colours on the sides are still mixed up - that is the next two stages.

6Permuting the corners

Goal: the corners on the top and on the bottom in the right order - on each side both corners have the same colour.

Look at the sides of the top layer. On each side there are two corners (and an edge between them - ignore it for now). If both corners on a side have the same colour, they are headlights - just like on the 3x3 cube. Count the sides with headlights:

• 4 - the corners of this layer are correct;
• 1 - two adjacent corners are swapped;
• 0 - diagonal corners are swapped.

Count the headlights on the bottom too. The algorithm / (0,3) / (3,3) / (0,3) / (6,6) swaps the two front corners on the top and the two front corners on the bottom at the same time (together with the edges next to them - you will solve those in the next stage).

1. One set of headlights on the top and one on the bottom: turn the top and the bottom so that the headlights are at the back, perform the algorithm. Done.
2. Any other case (a 0 or a 4 somewhere): take the layer that has one set of headlights and turn it so they are on a side - left or right (do not set up the other layer), and perform the algorithm. Now there is one set of headlights on the top and one on the bottom - as in point 1.

Finally, turn the top and the bottom by 90° (or 180°) so that the corner colours match the middle.

7Edges and parity

Goal: a solved Square-1.

The corners are in place, the edges are left. We do everything with one short algorithm B: (0,-1) / (0,-3) / (1,1) / (-1,2) / (0,1). It swaps the back and right edges of the top and the back and left edges of the bottom, nothing else.

Top: three edges in a cycle

Done twice with a top turn between them - B (3,0) B (-3,0) - it cycles three edges of the top: left, back and right. The front edge and the whole bottom stay put. It is like the “U perm” on the 3x3 cube:

1. Find an edge on the top that is in its place (its colour matches the corners next to it). Turn the top by 90° so that it is at the front.
2. Perform B (3,0) B (-3,0) and turn the top back. If the edges still do not match, do it once more.
3. No edge matches? Do it once from any position - one edge will start to match.

Bottom: the same

On the bottom it is the same, only between the two Bs you turn the bottom: B (0,3) B (0,-3). It cycles the left, back and right edges of the bottom; the correct edge should be at the front (turn the bottom, then turn it back).

Two swapped edges are left

Sometimes two edges on a layer match and two are swapped - you cannot fix that with a three-edge cycle. Check both layers:

• Two swapped on the top and two on the bottom: this is the easiest - a single B. Ideally set up the swapped top edges at the back and on the right, and the bottom ones at the back and on the left (turning the layers by 90° and then back). If the swapped edges are opposite each other, B still helps: then finish with a three-edge cycle.
• Only on one layer - this is parity. A Square-1 can have a single swap of two edges, because slices in odd shapes move pieces differently than cube moves do. You need a separate, long algorithm (below). It swaps the front and back edges of the top. If the swapped edges are opposite each other on the top - put them at the front and back, and you are done; if they are next to each other - perform it from any position and finish with a three-edge cycle. If the swap is on the bottom - perform it (the swap then appears on the top too) and do one B.

Parity comes up in roughly every second solve. Finally, turn the layers so that the colours match the middle. Solved!

Cheat sheet: all algorithms

Slice at the front, left of centre, white on top. All algorithms from stage 4 on keep the cube shape.

Printable cheat sheet (A4 / PDF)

1. Star - triple on the top: to the right, bottom with edges on the right, /. No triple: make one with a slice at the slice line (top-left + bottom-right on the other side), so that it can be sent down afterwards. Do not undo a slice.
2. Cube (edges between the corners on the top) - 0: / (4,2) / (1,2) / (3,3) / (-1,1); 1: / (0,2) / (1,2) / (-2,3) / (2,1) / (3,0) /; 2: / (4,2) / (1,-4) / (3,0) / (-1,1); 3: / (4,0) / (1,0) / (3,-2) / (2,1) / (0,3) /; 4: / (2,2) / (0,-1) / (3,3) /. Middle: / (6,0) / (6,0) / (6,0).
3. Corners to their layers - yellow corner at the front right on the top, white one at the front left on the bottom: (0,-1) / (0,3) / (0,1).
4. Edges to their layers - yellow on the right of the top, white on the right of the bottom: (1,0) / (0,-3) / (0,-3) / (-1,-1) / (1,4) / (0,3) / (-1,0).
5. Permuting the corners - / (0,3) / (3,3) / (0,3) / (6,6) swaps the front corners of the top and the bottom. One set of headlights on both: headlights at the back, once. Otherwise: one set of headlights on a side, once, and then as above.
6. Edges - B = (0,-1) / (0,-3) / (1,1) / (-1,2) / (0,1) (back+right of the top, back+left of the bottom). Three-edge cycle: top B (3,0) B (-3,0), bottom B (0,3) B (0,-3), correct edge at the front. Two swapped on both layers: B. On one layer: parity / (3,3) / (1,0) / (-2,-2) / (2,0) / (2,2) / (-1,0) / (-3,-3) / (-2,0) / (3,3) / (3,0) / (-1,-1) / (-3,0) / (1,1) / (-4,-3) (front-back of the top).

Solve it yourself

A simulator for practice. You can scramble the whole puzzle or pick a single stage - then you get a state in which the previous stages are already done. The slice gets blocked just like on a real Square-1 when a corner lies across the slice line. The buttons have the most common turns; a horizontal swipe on a layer turns it by one slot, and a vertical swipe on the right half does a slice.

Common problems

The slice will not go through

Some corner lies across the slice line - at the front or the back, on the top or the bottom. Turn that layer by one slot one way or the other. In an algorithm this almost always means that the previous turn was one slot too short or too long. Do not force it: that is an easy way to pop or damage a Square-1.

I keep going round in circles and have no star

Most often a slice undoes the previous one, because the layers went back to the same position between them (rule 3). The second reason is making a triple at the cost of two or three corners from the bottom - after such a join, sending the triple down only makes up for the loss. Choose joins that leave room on the bottom: only edges on the bottom right, or edges with a single corner.

The “cube” algorithm does not fit my case

Count the edges between the corners on the top along the shorter side (from 0 to 4) and set up the top exactly as in the animation: in cases 1 and 3 it matters which edge is on which side of the slice line. Always set up the bottom so that the slice passes between the corners of the star.

After an algorithm the puzzle is not a cube

From stage 4 on every algorithm keeps the cube shape, so something went wrong with the turns. The easiest thing to get wrong is the direction of the bottom: a positive number for the bottom is a turn like D on a cube - the front moves to the right. The second trap is holding it wrong: the narrower part of the middle must be at the front left. If the shape has fallen apart, go back to stage 2 - the colours you have already solved will need fixing anyway.

Two swapped edges are left at the end

That is parity, not a mistake - it comes up in roughly every second solve (stage 7). When both layers have two swapped edges each, a single B is enough.

The middle is misshapen

The right half of the middle is flipped. Align the layers and perform / (6,0) / (6,0) / (6,0) - nothing apart from the middle will change.

I turned the puzzle upside down

When it is upside down, the slice line on the front face is right of centre and all algorithms work as a mirror image. Turn the Square-1 back so that the narrower part of the middle is at the front left and the white layer is on top. In this course you never have to turn the puzzle over.

How long does it take to learn?

The cube shape usually “clicks” after a dozen or so attempts - practise it separately in the simulator. A full solve with this method takes about 60-70 slices; at first it takes 5-10 minutes, and after a week 2-3 minutes. A Square-1 is nice to solve when the slice goes through easily - see the Square-1 puzzles in our shop.

What next?

Once you can solve the Square-1 a few times without looking at the course, start with the shape: advanced solvers know the way to cube shape from each of the few dozen layer shapes and do it in at most 7 slices. Then the colours in fewer steps: corners and edges to their layers at the same time, and eventually the last stages with one algorithm per case.

The Square-1 is an official WCA competition event - the best solvers do it in a few seconds. Scrambles at competitions are written exactly as in this course, so with the course you can already scramble the puzzle using the official notation.