Megaminx parity
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I have an old 12-colored Megaminx that I put all new stickers on because the old ones were falling off. This Megaminx was in more of a state of disrepair than I originally thought, though, and when I was solving it 2 of the pieces (1 edge and 1 corner) popped out and fell on the floor. I wasn't paying attention to those particular pieces, so I had no idea which way they were facing when they popped out.
I plugged them back into the puzzle. I had no idea if they had the correct orientation or not though. Surprisingly, I was still able to complete the puzzle without disassembling it.
I know that a Rubik's Cube has parity; only $frac{1}{12}$ of the ways to assemble its cubelets results in solvable cubes. My intuition tells me that the Megaminx obeys the same parity rules (when I first got my Megaminx back in high school I was able to solve it using only the algorithms that come with a Rubik's Cube, with a few minor tweaks); however, I lack the mathematics background to verify this.
My question: If I were to completely disassemble a Megaminx and reassemble it at random, what are the odds that the resulting state would be solvable?
group-theory algorithms recreational-mathematics
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
$endgroup$
add a comment |
$begingroup$
I have an old 12-colored Megaminx that I put all new stickers on because the old ones were falling off. This Megaminx was in more of a state of disrepair than I originally thought, though, and when I was solving it 2 of the pieces (1 edge and 1 corner) popped out and fell on the floor. I wasn't paying attention to those particular pieces, so I had no idea which way they were facing when they popped out.
I plugged them back into the puzzle. I had no idea if they had the correct orientation or not though. Surprisingly, I was still able to complete the puzzle without disassembling it.
I know that a Rubik's Cube has parity; only $frac{1}{12}$ of the ways to assemble its cubelets results in solvable cubes. My intuition tells me that the Megaminx obeys the same parity rules (when I first got my Megaminx back in high school I was able to solve it using only the algorithms that come with a Rubik's Cube, with a few minor tweaks); however, I lack the mathematics background to verify this.
My question: If I were to completely disassemble a Megaminx and reassemble it at random, what are the odds that the resulting state would be solvable?
group-theory algorithms recreational-mathematics
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
$endgroup$
1
$begingroup$
You might enjoy David Joyner's book on Adventure's in Group Theory. It describes the group structure of the megaminx and confirms it has parity. For two other puzzles it compares the size of the (legal and illegal)-moves group to the (only legal)-moves group, but it didn't explicitly do it for the megaminx. For someone who was (or became) familiar with this sort of group theory, this would be a very easy question.
$endgroup$
– Jack Schmidt
Feb 1 '14 at 18:54
$begingroup$
Thanks for the tip! Does that book describe enough group theory for beginners that I'd be able to muddle through the problem myself?
$endgroup$
– Ryan Kennedy
Feb 1 '14 at 18:57
$begingroup$
It appears to be geared towards puzzle enthusiasts who have not yet taken discrete mathematics or abstract algebra (two math courses that are standard for CS majors usually in their 2nd or 3rd year of college).
$endgroup$
– Jack Schmidt
Feb 1 '14 at 19:04
add a comment |
$begingroup$
I have an old 12-colored Megaminx that I put all new stickers on because the old ones were falling off. This Megaminx was in more of a state of disrepair than I originally thought, though, and when I was solving it 2 of the pieces (1 edge and 1 corner) popped out and fell on the floor. I wasn't paying attention to those particular pieces, so I had no idea which way they were facing when they popped out.
I plugged them back into the puzzle. I had no idea if they had the correct orientation or not though. Surprisingly, I was still able to complete the puzzle without disassembling it.
I know that a Rubik's Cube has parity; only $frac{1}{12}$ of the ways to assemble its cubelets results in solvable cubes. My intuition tells me that the Megaminx obeys the same parity rules (when I first got my Megaminx back in high school I was able to solve it using only the algorithms that come with a Rubik's Cube, with a few minor tweaks); however, I lack the mathematics background to verify this.
My question: If I were to completely disassemble a Megaminx and reassemble it at random, what are the odds that the resulting state would be solvable?
group-theory algorithms recreational-mathematics
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
$endgroup$
I have an old 12-colored Megaminx that I put all new stickers on because the old ones were falling off. This Megaminx was in more of a state of disrepair than I originally thought, though, and when I was solving it 2 of the pieces (1 edge and 1 corner) popped out and fell on the floor. I wasn't paying attention to those particular pieces, so I had no idea which way they were facing when they popped out.
I plugged them back into the puzzle. I had no idea if they had the correct orientation or not though. Surprisingly, I was still able to complete the puzzle without disassembling it.
I know that a Rubik's Cube has parity; only $frac{1}{12}$ of the ways to assemble its cubelets results in solvable cubes. My intuition tells me that the Megaminx obeys the same parity rules (when I first got my Megaminx back in high school I was able to solve it using only the algorithms that come with a Rubik's Cube, with a few minor tweaks); however, I lack the mathematics background to verify this.
My question: If I were to completely disassemble a Megaminx and reassemble it at random, what are the odds that the resulting state would be solvable?
group-theory algorithms recreational-mathematics
group-theory algorithms recreational-mathematics
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
asked Feb 1 '14 at 17:17
Ryan KennedyRyan Kennedy
1506
1506
1
$begingroup$
You might enjoy David Joyner's book on Adventure's in Group Theory. It describes the group structure of the megaminx and confirms it has parity. For two other puzzles it compares the size of the (legal and illegal)-moves group to the (only legal)-moves group, but it didn't explicitly do it for the megaminx. For someone who was (or became) familiar with this sort of group theory, this would be a very easy question.
$endgroup$
– Jack Schmidt
Feb 1 '14 at 18:54
$begingroup$
Thanks for the tip! Does that book describe enough group theory for beginners that I'd be able to muddle through the problem myself?
$endgroup$
– Ryan Kennedy
Feb 1 '14 at 18:57
$begingroup$
It appears to be geared towards puzzle enthusiasts who have not yet taken discrete mathematics or abstract algebra (two math courses that are standard for CS majors usually in their 2nd or 3rd year of college).
$endgroup$
– Jack Schmidt
Feb 1 '14 at 19:04
add a comment |
1
$begingroup$
You might enjoy David Joyner's book on Adventure's in Group Theory. It describes the group structure of the megaminx and confirms it has parity. For two other puzzles it compares the size of the (legal and illegal)-moves group to the (only legal)-moves group, but it didn't explicitly do it for the megaminx. For someone who was (or became) familiar with this sort of group theory, this would be a very easy question.
$endgroup$
– Jack Schmidt
Feb 1 '14 at 18:54
$begingroup$
Thanks for the tip! Does that book describe enough group theory for beginners that I'd be able to muddle through the problem myself?
$endgroup$
– Ryan Kennedy
Feb 1 '14 at 18:57
$begingroup$
It appears to be geared towards puzzle enthusiasts who have not yet taken discrete mathematics or abstract algebra (two math courses that are standard for CS majors usually in their 2nd or 3rd year of college).
$endgroup$
– Jack Schmidt
Feb 1 '14 at 19:04
1
1
$begingroup$
You might enjoy David Joyner's book on Adventure's in Group Theory. It describes the group structure of the megaminx and confirms it has parity. For two other puzzles it compares the size of the (legal and illegal)-moves group to the (only legal)-moves group, but it didn't explicitly do it for the megaminx. For someone who was (or became) familiar with this sort of group theory, this would be a very easy question.
$endgroup$
– Jack Schmidt
Feb 1 '14 at 18:54
$begingroup$
You might enjoy David Joyner's book on Adventure's in Group Theory. It describes the group structure of the megaminx and confirms it has parity. For two other puzzles it compares the size of the (legal and illegal)-moves group to the (only legal)-moves group, but it didn't explicitly do it for the megaminx. For someone who was (or became) familiar with this sort of group theory, this would be a very easy question.
$endgroup$
– Jack Schmidt
Feb 1 '14 at 18:54
$begingroup$
Thanks for the tip! Does that book describe enough group theory for beginners that I'd be able to muddle through the problem myself?
$endgroup$
– Ryan Kennedy
Feb 1 '14 at 18:57
$begingroup$
Thanks for the tip! Does that book describe enough group theory for beginners that I'd be able to muddle through the problem myself?
$endgroup$
– Ryan Kennedy
Feb 1 '14 at 18:57
$begingroup$
It appears to be geared towards puzzle enthusiasts who have not yet taken discrete mathematics or abstract algebra (two math courses that are standard for CS majors usually in their 2nd or 3rd year of college).
$endgroup$
– Jack Schmidt
Feb 1 '14 at 19:04
$begingroup$
It appears to be geared towards puzzle enthusiasts who have not yet taken discrete mathematics or abstract algebra (two math courses that are standard for CS majors usually in their 2nd or 3rd year of college).
$endgroup$
– Jack Schmidt
Feb 1 '14 at 19:04
add a comment |
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$begingroup$
You might enjoy David Joyner's book on Adventure's in Group Theory. It describes the group structure of the megaminx and confirms it has parity. For two other puzzles it compares the size of the (legal and illegal)-moves group to the (only legal)-moves group, but it didn't explicitly do it for the megaminx. For someone who was (or became) familiar with this sort of group theory, this would be a very easy question.
$endgroup$
– Jack Schmidt
Feb 1 '14 at 18:54
$begingroup$
Thanks for the tip! Does that book describe enough group theory for beginners that I'd be able to muddle through the problem myself?
$endgroup$
– Ryan Kennedy
Feb 1 '14 at 18:57
$begingroup$
It appears to be geared towards puzzle enthusiasts who have not yet taken discrete mathematics or abstract algebra (two math courses that are standard for CS majors usually in their 2nd or 3rd year of college).
$endgroup$
– Jack Schmidt
Feb 1 '14 at 19:04