Do you have any ideas on how to check whether there is only one possible configuration with a given distribution of numbers? All I can think of is having to try every single combination of cubelets, save rotation of course, or to start off with one layer and then adding one more if the first is valid. But both ways seem to me like they'd take a lot of time to comb through.@Xias:
I am honestly still not sure I understand what you mean. If I got it right, you're talking about only getting the information and then having to make the triangles yourself, which would lead to these two solutions.
- Code: Select all
\ 3 ^ 3 /
\ / \ /
\2 1/1 1\1 4/
\ / \ /
\ 1 /
\ 2 ^ 2 /
\ / \ /
\3 1/1 1\1 4/
\ / \ /
\ 1 /
which depends on how you write the numbers on the 123 piece. If that is the case, I intend on writing the numbers on the cubes for my project, instead of just giving the numbers. If I got you wrong, then please let me know, I'd really like to understand this. Also, I'm not sure what you mean by each number being adjacent to its twin, which is the requirement for the puzzle I stated anyway.@Everybody:
I got a lot of inspiration from Xias' post at January 16th @6:34pm UTC. So since then I have been mixing the requirements for my own project with the puzzle I originally stated, as I wanted to find the solution for this puzzle in order to further work on the project. But I have decided to go another way after Xias' post. So because I am now also interested in finding the minimum number, and to avoid any confusion, I'll restate the original puzzle that this thread is for solving, and then I'll try and figure out the puzzle for my project myself, based on the information I gathered here, which I thank all of you for.
The idea is to take a 3x3x3 cube that has some way of obtaining unique configurations. I chose a Rubik's Cube, but it could be any 3x3x3 cube for that matter. And the parity issue of the Rubik's Cube is not important either, the only point is having a 3x3x3 cube with a known and unique configuration. And also, using the Rubik's Cube, the mechanics aren't important, for the purpose of this puzzle just think of it as 27 cubes that are the exact same size and shape.
So I have such a cube in my hand. I remember the exact configuration it was in, and I then put white stickers on all 54 visible faces so that by looking at it, it's not possible to figure out the configuration, you'd actually have to peel off some stickers but that's not allowed. Having done that, I then disassemble the cube into its 27 cubelets. All 54 faces that were hidden before on the inside are black, and I write a number, a letter or a symbol on them. It doesn't matter what it is, but let's go with numbers for this. Each black face HAS to have a number, and if two black faces met before the disassembly they get the same number. Thus it can be done with 27 pairs of numbers.
The idea is then to number the black face pairs in such a way that I can give the 27 cubelets to any person, even someone who doesn't know about Rubik's Cubes, and tell that person that their job is to assemble the cube back together, by pairing off the black faces with the same number on them and ending up with a 3x3x3 cube. When that person has done that, I then peel off the white stickers to ensure that he has done so correctly.
The idea of the puzzle is to find the minimum number of numbers/letters/symbols that can be applied to all black surfaces, but in such a way that it's only possible to reach ONE configuration. There might be 10 1s which is fine, as long as all but one way of pairing off those 1s and the rest of the numbers are invalid. If there's more than one way, this way of placing the numbers is not a valid solution.
And that's the description of the puzzle. The reason I mentioned blanks at one point is because I didn't want to put numbers on all black pairs. But as it has pointed out, the minimum number is just one less if you use blanks, as one of the numbers used in the solution can be replaced by a blank surface. So blanks are only important to my project.