# Instructions

Permutant is an iOS app that helps you learn to do the Rubik's Cube and related puzzles through a series of 26 challenges of increasing difficulty.

When you open the app, you will see the main menu

There are 26 challenges of increasing difficulty. You can choose to do them in order, or you can jump around the challenges, whatever way you want to enjoy the puzzles.

There are three categories of challenges:

- Challenges 1 to 12 are on a square grid
- Challenges 13 and 14 are on a triangle puzzle
- Challenges 15 to 26 are with a Rubik's Cube

Here are the instructions for each of the challenges.

Challenge 1

In this challenge there are 16 numbers in the 16 squares. The goal is to get the numbers to be in order from 1 to 16. The legal move you can make is that you can swap any two numbers by clicking on them. When you click the first number it will turn green. When you click the second number, it will turn green and then the two numbers will swap positions. In the screen shots below, the user has swapped the 4 with the 10.

Challenge 2

In this challenge, the user can select any three numbers. After the third number is selected, the number that was clicked first will move to the position of the number that was clicked second, the number that was clicked second will move to the position of the number that was clicked third, and the number that was clicked third will move to the position of the number that was clicked first. In the screen shots below, the user clicked on the numbers 2, 8, 4. The 2 moved to the position where the 8 was, the 8 moved to the position where the 4 was, and the 4 moved to the position where the 2 was.

Challenge 3

In this challenge, you use the rules from challenge 1 where you can swap any two numbers. Instead of being randomly shuffled, though, the only numbers that need to be fixed are three numbers that need to be 'cycled' like you did with challenge 2. The three numbers that need to be fixed are red. The goal is to find how to do a '3-cycle' by using only swaps.

Challenge 4

In this challenge, the numbers are all in their correct places except that there have been two swaps. In the screen shot below, the 13 and the 5 have been swapped and the 3 and the 8 have been swapped. The goal is to fix the numbers using the move from challenge 2. You are trying to accomplish two swaps by using 3-cycles.

Challenge 5

This is similar to challenge 1 where you can only do swaps of two numbers. But in this challenge, one of the numbers must be the number in the top left corner.

Challenge 6

All the numbers are in the correct position except the numbers 4, 9, and 10. The 10 needs to go to the position where the 9 is, the 9 needs to go to the position where the 4 is, and the 4 needs to go to the position where the 10 is. If you could do a 3-cycle move like in challenge 2, this could be accomplished in one move.

But for this challenge, there are only seven legal moves which can be done with the orange buttons at the top.

The first button says "A=(4 15)". If you click this button, it will swap the number that is in position 4 with the number that is in position 15. It will not swap the numbers 4 and 15.

Next to that button is one that says "A inv". This does the 'inverse' of A which 'undoes' it.

The last button says "D=(11 12 15) which makes a 3-cycle where the number in position 11 goes to position 12, the number in position 12 goes to position 15, and the number in position 15 goes to positiono 11.

The goal is to accomplish the (4 10 9) change by only using the legal moves.

Challenge 7

In this challenge there are only 9 legal moves. If you click on any of the numbers that are not on the right edge or the bottom edge, it will make an upside down 'L' shape and do a 3-cycle with those three numbers. In the screen shots below, the user clicked on the number 14 and it highlighted the 14, the 11, and the 15. Then it cycled the three numbers clockwise so the 14 went to where the 11 was, the 11 went to where the 15 was, and the 15 wend to where the 14 was. This is a challenging puzzle. If you don't get it, you still can continue to other puzzles.

Challenge 8

This is the classic 15 puzzle where there is an empty spot and you have up to 4 options each time to slide a tile that is either above, below, to the left, or to the right of the empty spot into the empty spot. In the screen shot below, the user has four options. Either the 5, the 1, the 13, or the 11 can be moved to the empty spot.

This is also a difficult challenge that has been around for over 100 years. This is easier, though, than challenge 7 since you do have a lot more moves available.

Challenge 9

This challenge demonstrates that if you do a permutation repeatedly, the board will eventually get back to its original state. The only move available button is "A=(2 4 11)(3 8)(6 9 12 14)". Any random shuffling can be thought of as a bunch of cycles. In this case when you click the button, the number in position 2 goes to position 4, the number in position 4 goes to position 11, and the number in position 11 goes to position 2. Also the numbers in positions 3 and 8 swap places. Also, the number in position 6 will go to position 9, the number in position 9 will go to position 12, the number in position 12 will go to position 14, and the number in position 14 will go to position 6. It looks like a random scrambling of the numbers, and doing it over and over might seem like it will make it even more mixed up. But that's not what happens. Doing it over and over makes it, at times, less shuffled. Play around with this and see if there is a way to predict when it will get back to the starting state. Also are there some other numbers of applications of this move that make it less mixed up than just doing it once?

Challenge 10

This one is just a fun puzzle where you can do one of four moves and their inverses. You can cycle all the numbers on the outside edge clockwise or counterclockwise, you can cycle the four interior numbers clockwise or counterclockwise, or you can move the numbers on either of the diagonals.

Challenge 11

Here, you have two different moves available and their inverses. Though the moves are somewhat complicated, it is possible to do a 3-cycle with only four moves.

Challenge 12

Using two fairly complicated moves and their inverses, it is possible to do the two 'swaps' with only four moves.

Challenge 13

The goal is to get the numbers so that the top number is 1, in the middle row the numbers are 2 and 3, and in the bottom row the numbers are 4, 5, and 6. By clicking on the three red circles, you can rotate three of the triangles. If you push the purple 'clockwise' button then it will change it so pushing a red button will make the triangles rotate counterclockwise. To get back to clockwise, press that button again. This can be accomplished in 4 or 5 moves.

Challenge 14

This has the same rules as challenge 13, but the numbers are more shuffled.

Challenge 15

In this challenge, the goal is to get the numbers 1 to 27 in their proper order. There are three things you are allowed to do.

1) You can do a 3-cycle with any three red numbers. When you select three red numbers, they will 3-cycle and the number cube at the top will make the moves to accomplish this.

2) You can do a 3-cycle with any three blue numbers. When you select three blue numbers, they will 3-cycle and the number cube at the top will make the moves to accomplish this.

3) You can press any of the 6 green numbers. This will cause an 8-cycle to happen. Moves 5 and 23 are the ones that are easiest to visualize when you see that the 6 green moves correspond to the six possible faces on a Rubik's Cube.

Challenge 16

This is the same as challenge 15 but with an added two moves. Some of the blue numbers will be upside down, some of the red numbers will be rotated 120 degrees counterclockwise and some red numbers will be rotated 120 degrees clockwise. The goal is to get the 27 numbers in order and to have them all right side up. To flip two blue numbers, press the blue button on the left and then select the two blue numbers you want to flip. For the red numbers, first press the red button on the left then select a number that needs to be rotated 120 degrees clockwise and then one that needs to be rotated 120 degrees counterclockwise (Think the first number has its bottom on the top right and the second has its bottom on the top left).

Challenge 17

Using three moves, try to get the 27 into its proper spot without disrupting the bottom layer. The numbers at the bottom correspond to the center sticker of each face. The top row of buttons is for clockwise and the bottom row is for counterclockwise. There is also a move with the number 14. The 14 is the center square of the entire cube, and can't be seen. This move moves the middle layer. It is not needed for this challenge. Also, there are five purple buttons on the bottom. The first four allow you to rotate the whole cube around. The fifth button restores the orientation of the cube to the original position.

It is a good idea to get an actual Rubik's cube and some post-it notes so you can try these challenges out on an actual cube. Having an actual cube will be useful for the rest of the challenges.

Challenge 18

This is like challenge 17. Three moves will get cube 26 into its proper place in the bottom layer.

Challenge 19

It takes 8 moves to move the cube in position 9 to position 27, the cube in position 27 to position 25, and the cube in position 25 to position 9. A hint is that the first three moves are the same as challenge 17 and the last three moves are the inverse of the first three moves.

Challenge 20

It takes 8 moves to move the cube in position 12 to position 26, the cube in position 26 to position 24, and the cube in position 24 to position 12. A hint is that the first three moves are the same as challenge 18 and the last three moves are the inverse of the first three moves.

Challenge 21

Using either 6 or 7 moves, get the make the cube with the red, white, and green sticker so that the white sticker is in the top face.

Challenge 22

Try to flip the cube with the white and green sticker so that the white sticker is on the top face. It can be done in about 10 moves.

Challenge 23

Using 13 or 15 moves, try flip the red, white, green clockwise and the brown, white, green counterclockwise. A hint is that the first 6 (or 7) moves are the same as challenge 21 and the last 6 (or 7) moves are the inverse of the first moves.

Challenge 24

Using about 20 moves, try flip both the white, green cube and the white red cube. The first moves are the same as in challenge 22 and the last moves are the inverse of the first moves.

Challenge 25

Click mix and then try to solve the number cube. Use the methods from challenge 19 and 20 to solve this the way you did challenge 15.

Challenge 26

Click mix and try to solve the Rubik's cube. Use the methods from challenges 19, 20, 23, and 24 to solve this the way you did challenge 16.