Alright, it turns out that the bruteforce passcode puzzle works in a kinda strange manner. Without experience, one would assume that all combinations are possible, but this is not the case, and here are the rules around what combinations are allowed. The general fact is that every stone has a placement in the sequence that it is not allowed to be in. If Stone B's placement is the placement that Stone A is not allowed to have, Stone A is also not allowed to have the placement that Stone B is never allowed to have.(PS: I have determined that the rule I just stated is an oversimplification of the actual rule, but it still covers every case for 4-stone and 5-stone codes, I will get into detail at the 7 stone section) Now with those general rules out of the way, let me state the specifics:
When there are 4 stones, they have a layout like:
2 4
1 3
Stone 2 is not allowed to be first.
Stone 1 is not allowed to be second.
Stone 4 is not allowed to be third.
Stone 3 is not allowed to be fourth.
Because of the second general rule, if 1 is the first stone in the sequence, 2 cannot be second. Similarly, if 2 is the fourth stone sequence, 3 is not allowed to be first. etc...
When there are 5 stones, they have a layout like:
1 3 5
2 4
Stone 2 can't be first
Stone 1 can't be second
Stone 4 can't be third
Stone 3 can't be fourth
Stone 5 can't be fifth
If you ignore the second general rule, it may seem like 13524 is possible; however, this is not the case because since 5 is in the third place, 4 cannot be in the fifth, so if the first three stones are 135, the final two are 42.
When there are 7 stones, they have a layout like:
0 2 4 6
1 3 5
Stone 2 can't be first
Stone 1 can't be second
Stone 4 can't be third
Stone 3 can't be fourth
Stone 5 can't be fifth
Stone 0 can't be sixth
Stone 6 can't be seventh
I called the first stone stone 0, because that would make it so that it is always the case regardless of how many stones there are that Stone 2 can't be first, Stone 1 can't be second, Stone 4 can't be third, and Stone 3 can't be fourth.
PS: The general rule actually is that there must be a full cycle where a stone is at a point in a sequence where another stone is and that stone is at a point in a sequence where another stone is and that looking at it like that passes through every stone and makes a full circle. An example of a sequence that is not possible because of this rule, but would be possible with the rules stated previously is 1306452 because 1 is where 2 can't be and 2 is where 6 can't be and 6 is where 3 can't be and 3 is where 1 can't be. This cycle doesn't go through 0, 4, or 5, so it isn't a possible pattern. This means that if you know the sequence starts with 13064, you actually can know that 25 is the end. In fact, in all cases, if there are only two stones left, you can logically determine which stone must go first.
Knowing what patterns are possible with 7 stones is mostly only really helpful if you take advantage of pausing the timer by, for example, losing focus on the game.
In conclusion, if all combinations were possible the amount of combinations for the amount of stones would be:
4: 24
5: 120
7: 5040
But instead the amount of actual possible combinations are:
4: 6
5: 24
7: 720 (it would be 1140 if the oversimplified general rules were all-encompassing)
Anyway, I found this behavior to be very odd, and knowing what sequences are possible helps a lot for 4 stones, a bit for 5 stones, and a bit for 7 stones depending on whether you take advantage of pauses or not, so it's nice to know.
PPS: I imagine that the game shuffles it like this:
Before shuffling, the stones have a starting arrangement:
A C B D E F A C G
B D A C B D E
Where A is meant to be clicked first, B is meant to be clicked second and so on. Then, a stone is removed and replaces another stone at random. For example, A might randomly replace D. Then, the replaced stone replaces a stone that has not already been replaced. For example, D might then replace C. This process is repeated until the final stone is moved, and it simply takes the empty space left over from the movement of the first stone with no other stones having not been replaced. An example of this in the 7 stone sequence may be: A replaces D, D replaces C, C replaces G, G replaces E, E replaces B, B replaces F, and F is placed where A started. The End result of this would be:
B F D C
E A G
where A is to be pressed first, B is to be pressed second, and so on. Using my other notation, the sequence in this example can be notated as 3064125.