Same problem, but your solution must be non-adaptive. IOW, specify the weighings that you will make up front. You are not allowed to change them based on the results of previous weighings
So the whole problem becomes: Identify the odd (heavy or light) ball among 12 identical balls with a static plan (i.e. no conditional decisions) that uses at most 3 weighings of a balance scale.
The original puzzle is hard enough, but this variant has been killing me. I think I could come up with a solution if the original problem specified whether we expect the odd ball to be heavy or light.
Anyone have any ideas? It feels nearly impossible.