I challenged myself to create a card game that simulates the experience of solving a 3x3 Rubik's cube. I have a first prototype but I'm now stuck at knowing if every random initial state will be solvable.
I will first explain the challenge I'm facing and after that I will provide more context of the actual game, but if you want to check the cards and/or rules first, you can do it here.
In a 3x3 Rubik's cube not every state is solvable, we need to maintain the parity. Of course, if we mix the cube using only legal movements, we are always going to have a solvable state. This is where the first particularity of my game comes in:
- My idea is to simplify the set-up of the game by shuffling the 9 cards together and place them in the table in a 3x3 grid.
Problem: When shuffling the cards I don't know if I'm performing any illegal movements that will generate an unsolvable state.
With that challenge in mind I will proceed to explain the game:
- The game consists of 9 identical double-sided cards but each card having a different number [1-9]. See this image of a single card showing both sides. The dots that appear in the card where the arrows are pointing are just for knowing which color is on the other sides without looking.
- To set up the game the cards will be shuffled, flipped, and rotated so we have a random initial state. Then, they will be placed in a 3x3 grid on the table making sure each card covers the bottom half of the card in the previous row. See this example of a set up game (Notice we cover the bottom half of every card, except for the cards in the last row as we don't have more cards to cover them).
- Now, the actions we can do for solving the game are:
- Shift rows: For shifting a row, we move the 3 cards of the selected row one step to the right or left. There will be one card that goes off the grid. That card is then flipped horizontally and placed at the opposite side of the grid, where there should be an empty cell. See the following example. Notice the horizontal flip in Step 3
- Shift columns: Shifting columns is similar to shifting rows. We move the complete column one step up or down. The difference is that the card that goes off the grid has to be flipped vertically See the following example. Notice the vertical flip in Step 3
The rule of thumb is the card that goes off the grid will be flipped in the direction of the shift and placed at the opposite side of the grid
- Goal: The puzzle will be completed when all the 9 cards are in numerically ascending order (from left to right and top to bottom), and with the green faces at the top. See example of a solved puzzle
I hope I explained that well, let me know if you want me to clarify something :D
You can download the Print and Play files of the cards and the rules with more detailed explanation here.
- Will shuffling the cards always generate a solvable state?
- Do you think there might be a golden path to easily solve the puzzle every time?
- How many possible combinations the puzzle has? My guess is that it has 95,126,814,720 possible states (counting the solved state). It will be awesome if all of them are solvable :D
- Would the same scenario apply to a 2x2 puzzle and a 4x4 puzzle?