Mathematics and intuition

We all make a number of decisions based on our intuition. We feel like it, and so do it. Is there any logic behind those feelings? There must be, because all of us want to maximize our own chances, our development, our pleasures and our evolution. So these choices can't be arbitrary, rather should have a rational basis, even though seeming intuitive. I shall try to make an analogy here for a case, with our intuition for it, and the seeming logic behind it.

Suppose there are are two sets of people. We have to group these people into pairs, so that they achieve a property called "maximum". This maximum can be of any thing, but we would like this maximum it to be driven towards maximum "goodness", "efficiency", etc. How do we group these people to achieve it? Do we group like minded people alike, or do we group unequal people in the pair? I am sure that most of the people would intuitively feel that grouping like minded people will help us achieving maximum harmony, co-operation, efficiency. It can be argued that grouping unlike minded people can create new ideas, creativity, etc etc, but on an average, this will not be the case. People will seek to be with like minded people for most of the time, and they will feel it intuitively.

Is there some logic or mathematical argument to support this? I shall present another example, and try to find the similarities between the above example and this one.

Let us have two identical set of numbers. 0Let us have say three numbers in each set. Now we want to pair the numbers, each from one set. We then find the sum of the product of all the pairs. What pairing will yield the maximum product for these set of numbers? Actually it is a well known result that if we pair equal numbers together, then their product will yield the maximum sum, than the sum of pairs with unequal numbers. It shows that being with "like", yields the property of maximum sum.

(a,b,c) (a,b,c)

Pairs possible:
1. Pairs with same numbers (a,a), (b,b), (c,c) : sum of pairs is a²+b²+c²
2. Pairs with unequal numbers (a,b), (b,c), (c,a) : sum of pairs is ab+bc+ca

And it is well known that a²+b²+c² > ab+bc+ca

It raised some question in my mind. Can this inequality be taken as a basis to say that equal pairing is more efficient, that what we feel intuitively. Or is it just reading between the lines? But this does surprise me that we live in a world, where if we make the pair of equal numbers, it gives the maximum product. This product can be treated as an interaction between two objects. This interaction yields maximum result when they are equal. I do think there are similarities between this multiplication and the human interaction, and thats why we like to go to same-minded fellows.

So guys, feel more confident when you next time go to make a pair, if you are confused over whether to go for same mindset or interacting with people with different tastes. Diversity is not always good, at least on an average. And now you do have a reason for that.