3 ms·

"Four beer caps are placed on the corners of a square table with arbitrary orientations...." Please can you tell us the solution to this puzzle? I cannot find t

"Four beer caps are placed on the corners of a square table with arbitrary orientations...." Please can you tell us the solution to this puzzle? I cannot find the answer anywhere online. What is the solution?

NoodleIncident 12y ago
Start from the last step: AB BA From here, flipping two opposit corners will result in all 4 being the same way. One step before: AA BB Flipping any two adjacent ones will either solve it or get you to the previous step. So from here, we can see that any state with two up and two down is solvable. The state with all four one way or the other is already solved. The only thing left is the state with just one out of line. AB BB No matter which individual one is flipped, you are in a solvable state.
archgoon 12y ago
Suppose all caps face the same direction... DONE! Suppose two caps (AB) face the same direction, if they're diagonal, then you can flip the diagonal. Regardless of which caps flip, all caps will then have the same orientation. If the two caps are along a side, then you perform a side flip. If you get the other two caps, or you flip AB, then all caps have the same orientation. If you flip one of the AB caps and one other, you now have a diagonal setup, which we have already solved. If you have one cap, then flip a cap, you will either have two caps in the same orientation (which we know how to solve) or all caps in the same orientation. Three caps is the same as one cap in the same orientation.
cturner 12y ago
There's other answers here that suggest it's enough to merely pass though a correct state, without having executed all of the operations. If that's valid, then the wording of the puzzle should state this more forcefully. Even then, the last sentence can't be satisfied. Imagine if you were competing against the machine, it could simply always select moves that left one corner in the down position. From my reading of this puzzle, you can't sample the board in order to capture the orientation of the caps. Hence, I don't believe there's a way you could reach an outcome. Also, the base case doesn't work. Imagine there was one cap. You can choose to flip it or not flip it. Can you get everything in the up position without sampling the board? "Can you devise a sequence that ensures they all face up? Down?" No. Garbage in, garbage out. The only thing I can think of that might be useful at the moment: repeatedly hammering some combination of sides and diagonals, causing some statistical pattern to emerge. But without knowing the origin states, not sure how to use that. There's a similar problem here: http://gurmeet.net/puzzles/tumblers-on-a-rotating-table/ http://gurmeet.net/puzzles/tumblers-on-a-rotating-table/
tunesmith 12y ago
I think the trick is that it's ok if the sequence makes them all the same orientation partway through the sequence.

NoodleIncident 12y ago

Start from the last step: AB BA From here, flipping two opposit corners will result in all 4 being the same way. One step before: AA BB Flipping any two adjacent ones will either solve it or get you to the previous step. So from here, we can see that any state with two up and two down is solvable. The state with all four one way or the other is already solved. The only thing left is the state with just one out of line. AB BB No matter which individual one is flipped, you are in a solvable state.

archgoon 12y ago

Suppose all caps face the same direction... DONE! Suppose two caps (AB) face the same direction, if they're diagonal, then you can flip the diagonal. Regardless of which caps flip, all caps will then have the same orientation. If the two caps are along a side, then you perform a side flip. If you get the other two caps, or you flip AB, then all caps have the same orientation. If you flip one of the AB caps and one other, you now have a diagonal setup, which we have already solved. If you have one cap, then flip a cap, you will either have two caps in the same orientation (which we know how to solve) or all caps in the same orientation. Three caps is the same as one cap in the same orientation.

cturner 12y ago

There's other answers here that suggest it's enough to merely pass though a correct state, without having executed all of the operations. If that's valid, then the wording of the puzzle should state this more forcefully. Even then, the last sentence can't be satisfied. Imagine if you were competing against the machine, it could simply always select moves that left one corner in the down position. From my reading of this puzzle, you can't sample the board in order to capture the orientation of the caps. Hence, I don't believe there's a way you could reach an outcome. Also, the base case doesn't work. Imagine there was one cap. You can choose to flip it or not flip it. Can you get everything in the up position without sampling the board? "Can you devise a sequence that ensures they all face up? Down?" No. Garbage in, garbage out. The only thing I can think of that might be useful at the moment: repeatedly hammering some combination of sides and diagonals, causing some statistical pattern to emerge. But without knowing the origin states, not sure how to use that. There's a similar problem here: http://gurmeet.net/puzzles/tumblers-on-a-rotating-table/ http://gurmeet.net/puzzles/tumblers-on-a-rotating-table/

tunesmith 12y ago

I think the trick is that it's ok if the sequence makes them all the same orientation partway through the sequence.