I think my logic for this q is correct, as I basically did what the ms did. I just treated 2 of the tiles (as a pair labelled below) as its own shape, and said that shape was rotated by B-a for each rotation, and you need to do that 27 times to get 360 degrees, so then as each shape is made from 2 tiles, you need 54 tiles.
But I had a random dumb thought after looking at the ms - why don't you multiply 27 by 3 because the ms's diagram has 3 tiles, and I haven't really been able to come to a satisfying answer - obviously you need to multiply by 2 when you treat 2 tiles as its own shape and say that shape has done the rotation, but why here do you not multiply by 3, since it requires 3 tiles for that rotation?
If anyone could explain it would be greatly appreciated.
Hmm... I'm a bit confused by what you mean by "requiring 3 tiles for that rotation". A picture would be helpful to illustrate your point.
But I think your confusion comes from the MS tricking you to look at 3 tiles only. The intuitive way is to continue the pattern, so imagine a 4th tile continuing "upwards" (hence completing the second pair of tiles, agreeing with your line of attack).
I think my logic for this q is correct, as I basically did what the ms did. I just treated 2 of the tiles (as a pair labelled below) as its own shape, and said that shape was rotated by B-a for each rotation, and you need to do that 27 times to get 360 degrees, so then as each shape is made from 2 tiles, you need 54 tiles. But I had a random dumb thought after looking at the ms - why don't you multiply 27 by 3 because the ms's diagram has 3 tiles, and I haven't really been able to come to a satisfying answer - obviously you need to multiply by 2 when you treat 2 tiles as its own shape and say that shape has done the rotation, but why here do you not multiply by 3, since it requires 3 tiles for that rotation? If anyone could explain it would be greatly appreciated.
Similar to tonys reply, the third tile just shows that it/the next pair is offset by a rotation beta-alpha. So double the amount of beta-alphas you get.
Is the right method of thinking to think about 2 of the tiles as pairs, and then each pair gets rotated by beta-alpha?, because I can't really see any other way.
Looking at the image with 3 or 4 tiles doesn't really make sense to me either because you multiply by 2 not 3 or 4, but the ms looks at each tile individually rather than them as pairs?
Or is that sort of such an obvious perspective that the ms doesnt write about it?
Is the right method of thinking to think about 2 of the tiles as pairs, and then each pair gets rotated by beta-alpha?, because I can't really see any other way. Looking at the image with 3 or 4 tiles doesn't really make sense to me either because you multiply by 2 not 3 or 4, but the ms looks at each tile individually rather than them as pairs? Or is that sort of such an obvious perspective that the ms doesnt write about it?
Yes, the diagram shows that the third tile (next pair) is rotated by beta-alpha. So double the number of beta-alphas in 360.
Is the right method of thinking to think about 2 of the tiles as pairs, and then each pair gets rotated by beta-alpha?, because I can't really see any other way. Looking at the image with 3 or 4 tiles doesn't really make sense to me either because you multiply by 2 not 3 or 4, but the ms looks at each tile individually rather than them as pairs? Or is that sort of such an obvious perspective that the ms doesnt write about it?
The point is that the pattern continues indefinitely (well, until it loops back around). The MS only shows 3 tiles in the infinitely long string of tiles to save paper space. Perhaps MS could be clearer by saying "consider pairs of tiles" somewhere in the beginning.
(Meta-commentary: Deciphering MS is often times a completely different problem than solving the problem itself. For me reading MS fully is a waste of energy, but to each of their own, I'd suppose.)
Now can you group 3 tiles as one group, and do the same calculation? Sure, but the angle of rotation (beta-alpha) would be different.