CaptainEd

I guess neither of us knows which objects have property M, and we rely on a third party to give us a determination at the end of each N phase or the STOP event?

So Plasmid's beautiful algorithm guarantees a maximum number of moves when the weights, expressed in an odd ternary notation, are RLLLLR That is, the 243 and the 1 are in the Right pan, while the U and all other weights are in the Left pan. In this case, there are 4 triple moves (move from Left to right, remove same weight, move same weight back to Left) and 2 single moves (Left to Right). Actually, since Plasmid starts with RLLLLL, there's only one single move. So this algorithm has a maximum number of moves of 3n-5, where n is the number of weights. The algorithm is so clear, and solves such an important problem, I'm happy to declare that as a solution. So, my next question is: what is the algorithm and max moves from a LeMans start--empty pans?

Plasmid, that is an elegant answer! Assuming Philip will allow us to preset all the weights before the first move. If not, you're up to 17. And you've proved it!

Looking at SP's answer, I think we are working on different problems. I think I am trying to find the minimax number of moves needed to find any given unknown weight. "minimax" means we need to minimize the maximum over all weights.

Are you sure the JOIN doesn't preserve the center of mass? Thank you for the statement that the SPLIT is not the inverse of JOIN--it really does look like SPLIT; JOIN does move the COM away from the origin. Thank you for a very tough problem. My reticence in the last week has not been due to lack of interest, but lack of progress...

I'd like to add that this whole area is bizarre, and yet it does seem (you've said so, and SP has shown an example) that one could do some sort of analysis for problems in this area. Thank you, James33, for a glimpse into a weird world!

I'm more accustomed to [-4.3] = (-5) Is that what you mean? Or do you really intend to take the absolute value of the floor?

I understand the die tipping over onto an adjacent face. Is that the "1/4 turn" you mention, or is there an additional quarter turn (ie, on some other axis)

(I"m wrong again, but the split appears to allow the COG to move inward...)

(Answer removed, had not noticed that (d > 0). I knew it was too easy!)

Oops, you're right, I got it backward.

Interesting! I got the same answer...,

Do we suppose he means that we don't have to reorder AFTER it reads BARBAPAPA? or we shouldn't reorder it from the OP?

Oh, how cute! Good job, Depew! This is an odd genre of a puzzle, Bonanova. I look forward to the next one. MrZincs, I have no idea how you arrived at that answer. Impressive job, all of you.

Yes, thanks. What a marvelous solution!

EventHorizon, I'm not grasping the method. In particular, I can't tell if the "group of three (or more)" is always at least three, or if it resets upon color change. In particular, how would you point the backs for the opening sequence, and what would suits would the observer call? SHDCSDHCSCHD

Haven't got a good solution yet. Partly because swapping the first two waypoints makes the first three legs dominated by 26, 26, and -60. I like the fact that both the X and the Z travel 26 between the first and second waypoints. I can't tell yet whether it's good to have long legs (like 60) (because the number of steps is roughly the square root of the distance), or it's bad (because it's long anyway).

Sorry to be vague. This graph shows the sequences of x-, y-, and z-coordinates. To get to the next waypoint quickest, you want to accelerate halfway and then decelerate. The path will look like a smooth parabola, as you see in most of the graph. When I started, with a solution having 4 more moves, one or two of the parabolas were not perfect. That gave me a hint of where to look for potential improvement. The words "I think..." were intended to convey that I am not certain that the graphing has actually helped.

Yes, that shows that it's a good puzzle--it was clearly stated, but still people got different answers. A lot of us learned something here.