Teaching my grandson about combinations. He has the following “Morphing Fidget Spinner”. Each of the six arms has six knuckles, each with 5 positions. So for each arm it is 5 to the sixth power = 15,625. So for the entire six arms is it just 15,625 x 6 = 93,750?
You have 36 knuckles. Each has 5 positions. That 5^36 ~= 14.5E24.
That assumes the 6 arms are distinguishable. Like if each was a different color or had a different tip or something.
But if the arms are indistinguishable then you have a decision to make. If you set a random configuration, snapshot it, then set the same configuration, but with each arm pattern set e.g. one arm clockwise from the first snapshot, is that the same configuration or a different one? Said another way, since the overall spinner has rotational symmetry in 6 increments, are you counting rotations of the same pattern as distinct or not?
If distinct, my answer stands. If not, divide by 6.
Either way, be careful asking a kid prone to obsessive behavior to try setting every one of the positions.You might just ruin the rest of his life.
@LSLGuy’s answer is also 15625^6.
Unfortunately, some of the positions are physically impossible, as the arms will bump into each other (or the body). So the real number will be less than 14.5e24. Possibly by a few orders of magnitude. It’s tricky to say how much, though.
Additionally you can flip it over and get the same configuration.
It’s more complicated than that, I’m afraid, because some configurations are themselves symmetric.
Good point. The divide by 6 accounts for one type of symmetry, but as you say there are many others.
Looking at the link in the OP, the central hub, at least for the first three sample images shown, is not symmetric. For the first 2 samples with the blue joints in the arms the little red crescents between the red dots on the hub are different sizes and two are missing. Likewise for the third sample with the red joints, again the little gold crescents between the gold dots on the hub are different sizes and three are missing. The resolution of the images for the last 2 samples is too low to see details on the central hub. Assuming that these samples are representative of the toy, then the orientation of the central hub, and thus of the arms, will always be distinguishable and so the total number of positions is (5^6)^6 or 5^36.
As @Dr.Strangelove mentions, some of the positions are not possible because the arms will be blocked by either their neighbours or the central hub. It should be possible to enumerate these physically impossible arrangements and so find the actual total number of positions, but it would help to actually have one to find this out. For example, it looks like it would not be possible to have any 2 adjacent arms with the second joints bent at 90 degrees towards each other whenever the first two joints are in the same orientation. There are 6 ways to choose 2 adjacent arms, there are 5 identical first joint positions, there is 1 physically impossible arrangement for the second joint, then (possibly) 5 possible positions for each of the remaining 4 joints per arm, so there are 6 x 5 x 2 x 5^4 = 37500 impossible arrangements that need to be subtracted from the total number.
Only if the underside of the hub is indistinguishable from the top, unless I’m not understanding what you mean by flipping it over.
There are a lot of impossible combinations. I started out to count them but lost interest. There are nearly as many impossible combinations as there are possible ones. If the grandson brings the toy back this weekend I’ll try to make a better count.