I saw this qustion:
Suppose you want to create a set of weights so that any object with an integer weight from 1 to 40 pounds can be balanced on a two-sided scale by placing a certain combination of these weights onto that scale.
What is the fewest number of weights you need, and what are their weights?
now the answer was weights of mass in powers of 3 (1,3,9,27, etc.), since they can be placed in either pan like... apparently this is not really what I conceive of as base-3, but something to the effect of a "-1,0,1" and not 0,1,2...
anyway, my question: what kind of efficient multiples of weight suspended from vertices would be required to balance a regular triangle... and then what kinds of multiples of weights would be required to balance a regular tetrahedron? I am assuming be began as with the scale problem with 1 (or maybe 2 or even 3 in the case of a tetrahedra) of the masses unknown and masses can be placed on any of the vertices to balance it out... presumably in the case of a tetrahedral the answer would vary based on how many vertices had unknown masses attached to them (I am guessing 1 or 3 masses is related to the 4 vertices/faces vs 2 is related to the 6 edges and may have an octahedral type symmetry).
can someone refer me to a good resource? barycentric coordinates seems to only apply in affine space while this seems to be a different sort of problem.
my guess would be that for a triangle the answer is 5^n: 1, 5, 25, 125, etc... no guess for the tetrahedral case except that maybe if there is only an unknown weight on 1 or 3 of the vertices that maybe it could be 7^n and possibly 13^n for the case of 2 unknown vertices