The problem is unclear. I think you have a labelled graph G=(V, E) with labels c:V->R, such that each node in V consists of a triple (L, R, S) where L is a sequence of weights are on the left, R is a sequence of weights that are on the right, and S is a set of weight that have been taken off. Define c(L, R, S) to be the centre of mass. Introduce an undirected edge e={(L, R, S), (L', R', S')} between (L, R, S) and (L', R', S') either if (i) (L', R', S') results from taking the first weight off L and adding it to S, or (ii) (L', R', S') results from taking the first weight off R and adding it to S, or (iii) (L', R', S') results from taking a weight from W and adding it to L, or (iv) (L', R', S') results from taking a weight from W and adding it to R.

There is a starting node (L_0, R_0, {}) and an ending node ({}, {}, W) , with the latter having L=R={}.

I think you're trying to find the path (L_n, R_n, S_n) from the starting node to the ending node that minimises the maximum absolute value of c(L_n, R_n, S_n).

I won't post a solution, as requested.

That problem is not clearly stated, so if you’re pasting that into an AI verbatim you won’t get the answer you’re looking for.

My guess is: first move the weights to the middle, and only then remove them.

However “weights” and “bar” might confuse both machines and people into thinking that this is related to weight lifting, where there’s two stops on the bar preventing the weights from being moved to the middle.

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