we were given a fixed number from which we can estimate
We were not given any such thing, you have incorrectly inferred a calculation as a possible solution to reveal an n.
We are given an Odds Ratio result that complies with a certain threshold change from baseline cognitive score, not the n counting how many patients met that threshold.
That n cannot be inferred as you attempt from a finger in the air placebo response estimate, but that n is crucial to understand if that n was say 5 or 76.