I have recently started to spend time on http://www.puzzlopia.com/ (EDIT: Site is now dead, see here: https://web.archive.org/web/20161007014010/http://www.puzzlopia.com/), which is a puzzle site. If you are not logged in, you are greeted with a "gear turning" puzzle, where you want all the "gears" oriented the same way.

[![Gear Puzzle][1]][1]

If, for whatever reason, you don't want to try it right now, I will describe it. The goal is to make all 9 gears oriented in the same direction (Those "dots" resemble the goal orientation). You can rotate all gears at the same time, 120 degrees in the same direction. However, you may also "lock" some gears, so they will not be affected while you orient the rest. Equivalenty, one may note that this is equivalent to rotating the locked gears in the oppoosite direction.

[![Rotate ex.][2]][2]

Here, I have locked the upper left gear. This consequently locks the two gears it's "pointing" at, to it's upper right and lower right.

Often, I have managed to orient all gears correctly, with the exception of one rebellious gear that won't comply. Trying to deliberately screw up the rest of the puzzle to tinker with the "dark forces" that prevent me from completing the puzzle, did not seem to help. After restarting about 5 times, I solved the puzzle in very few moves (The puzzle is randomly generated).

So, this is really interesting. Is this puzzle always solvable, no matter the starting orientations of the gears? Is there an algorithm to rotate just one specific gear? If it is not always solvable, what conditions are required to make it solvable? I suspect the dark magic of modular arithmetic is involved (I've never been good with that stuff).

(For those interested in wasting even more time in life, I'll note that I have found this site very enjoyable) [1]: https://i.sstatic.net/1bByK.png [2]: https://i.sstatic.net/q7Ef6.png

I have recently started to spend time on http://www.puzzlopia.com/ (EDIT: Site is now dead, see here: https://web.archive.org/web/20161007014010/http://www.puzzlopia.com/), which is a puzzle site. If you are not logged in, you are greeted with a "gear turning" puzzle, where you want all the "gears" oriented the same way.

If, for whatever reason, you don't want to try it right now, I will describe it. The goal is to make all 9 gears oriented in the same direction (Those "dots" resemble the goal orientation). You can rotate all gears at the same time, 120 degrees in the same direction. However, you may also "lock" some gears, so they will not be affected while you orient the rest. Equivalenty, one may note that this is equivalent to rotating the locked gears in the oppoosite direction.

Here, I have locked the upper left gear. This consequently locks the two gears it's "pointing" at, to it's upper right and lower right.

Often, I have managed to orient all gears correctly, with the exception of one rebellious gear that won't comply. Trying to deliberately screw up the rest of the puzzle to tinker with the "dark forces" that prevent me from completing the puzzle, did not seem to help. After restarting about 5 times, I solved the puzzle in very few moves (The puzzle is randomly generated).

So, this is really interesting. Is this puzzle always solvable, no matter the starting orientations of the gears? Is there an algorithm to rotate just one specific gear? If it is not always solvable, what conditions are required to make it solvable? I suspect the dark magic of modular arithmetic is involved (I've never been good with that stuff).

(For those interested in wasting even more time in life, I'll note that I have found this site very enjoyable)

I have recently started to spend time on http://www.puzzlopia.com/, which is a puzzle site. If you are not logged in, you are greeted with a "gear turning" puzzle, where you want all the "gears" oriented the same way.

[![Gear Puzzle][1]][1]

If, for whatever reason, you don't want to try it right now, I will describe it. The goal is to make all 9 gears oriented in the same direction (Those "dots" resemble the goal orientation). You can rotate all gears at the same time, 120 degrees in the same direction. However, you may also "lock" some gears, so they will not be affected while you orient the rest. Equivalenty, one may note that this is equivalent to rotating the locked gears in the oppoosite direction.

[![Rotate ex.][2]][2]

Here, I have locked the upper left gear. This consequently locks the two gears it's "pointing" at, to it's upper right and lower right.

Often, I have managed to orient all gears correctly, with the exception of one rebellious gear that won't comply. Trying to deliberately screw up the rest of the puzzle to tinker with the "dark forces" that prevent me from completing the puzzle, did not seem to help. After restarting about 5 times, I solved the puzzle in very few moves (The puzzle is randomly generated).

So, this is really interesting. Is this puzzle always solvable, no matter the starting orientations of the gears? Is there an algorithm to rotate just one specific gear? If it is not always solvable, what conditions are required to make it solvable? I suspect the dark magic of modular arithmetic is involved (I've never been good with that stuff).

(For those interested in wasting even more time in life, I'll note that I have found this site very enjoyable) [1]: https://i.sstatic.net/1bByK.png [2]: https://i.sstatic.net/q7Ef6.png

I have recently started to spend time on http://www.puzzlopia.com/ (EDIT: Site is now dead, see here: https://web.archive.org/web/20161007014010/http://www.puzzlopia.com/), which is a puzzle site. If you are not logged in, you are greeted with a "gear turning" puzzle, where you want all the "gears" oriented the same way.

[![Gear Puzzle][1]][1]

If, for whatever reason, you don't want to try it right now, I will describe it. The goal is to make all 9 gears oriented in the same direction (Those "dots" resemble the goal orientation). You can rotate all gears at the same time, 120 degrees in the same direction. However, you may also "lock" some gears, so they will not be affected while you orient the rest. Equivalenty, one may note that this is equivalent to rotating the locked gears in the oppoosite direction.

[![Rotate ex.][2]][2]

Here, I have locked the upper left gear. This consequently locks the two gears it's "pointing" at, to it's upper right and lower right.

Often, I have managed to orient all gears correctly, with the exception of one rebellious gear that won't comply. Trying to deliberately screw up the rest of the puzzle to tinker with the "dark forces" that prevent me from completing the puzzle, did not seem to help. After restarting about 5 times, I solved the puzzle in very few moves (The puzzle is randomly generated).

So, this is really interesting. Is this puzzle always solvable, no matter the starting orientations of the gears? Is there an algorithm to rotate just one specific gear? If it is not always solvable, what conditions are required to make it solvable? I suspect the dark magic of modular arithmetic is involved (I've never been good with that stuff).

(For those interested in wasting even more time in life, I'll note that I have found this site very enjoyable) [1]: https://i.sstatic.net/1bByK.png [2]: https://i.sstatic.net/q7Ef6.png