Find all natural numbers which are the cardinal of a set of nonzero Euclidean vectors whose sum is $ 0, $ the sum of any two of them is nonzero, and their magnitudes are equal.
all numbers beside 0(obviously) 1 2 and 4,that is because if the number isn't a power of 2,the number can be split into a number of regular polygons with the same ODD NUMBER OF SIDES as well as same SIDE LENGTH that are all slightly rotated so that we avoid cases in which we have parrallel vectors.If the number is a power of 2 greater than 4 than we create a triangle and with the remaining n-3(which is not a power of 2 since n>4) we do the previously mentioned process.
either the lenghts themselves or the distance from each point to the centroid of their respective polygon works