How many moves are at least needed to find a ball that is from the majority? We have $7$ balls that at least $4$ of them are made from iron.In a step we can give two of them to a person that can recognize whether  or not the balls are made from the same materials. How many moves are at least needed to find a ball that is from the majority?
My attempt:It is easy to see $4$ moves are enough it is just needed to divide the balls into $3$ sets of two balls and a single ball the check that three sets of balls.Then we have to divide the cases according to the number of "Yes" or "No"'s that we hear.Finally you can get in some cases three is enough and in some $4$ is enough.But I can't prove $3$ steps can't be enough. 
 A: We suppose you have a strategy.  I will be the person who can recognize the balls.  What I will do is I will tell you what my answers are (based on your strategy).  Then I will show that (a) there is an assignment of materials to the balls so that my answers are truthful and (b) you can't identify any iron ball after three questions.
If you ask me about two balls neither of which you have asked about before, I will say they are the same.  If you ask me about two balls at least one of which you have asked me about before, I will say they are different (except if you could already deduce what the correct answer was, in which case I give that answer, but it would be silly for you to ask such a question, because it's a wasted question).
Now, let's draw a graph whose vertices are the seven balls, and whose edges are the three pairs of balls you asked about.  For each of the connected components other than the isolated vertices, based on my answers, you know how the vertices are divided within that component into two materials.  These components have two, three, or four vertices (because there are only three edges).  Because of how I answered the questions, for each component, one material accounts for exactly two balls.    
We split into cases: either there is one component of size 4 split 2 and 2, together with 3 isolated vertices; or there is a component of size 3 split 2 and 1, a component of size 2 with both vertices the same, and 2 isolated vertices; or three components of size 2, each the same, and one isolated vertex.  In each of these cases, you don't have enough information to determine an iron ball, and clearly, my answers were correct for some assignment of materials to balls.  
