How good is your mathematical intuition?

There is solid evidence that we are much better at reasoning in context than in the abstract. I have come up with a simple test of this notion. Consider the following problems.

1. Given a small group of numbers, what would be a simple way of testing that no number is more than 5 times greater than any other number, using a minimum amount of calculation.

2. Now let me contextualize it. Suppose you are among a group of people wearing tags giving their ages. How would you go about checking that nobody is more than 5 times as old as anyone else?

Do you find the solution to the second problem to be more obvious? How would you put into words why your test works? I was not able to find a simple way of putting it into words, but I can prove that the method works with a simple algebraic inequality.

3. Given a small group of numbers, how would you go about showing that no number is greater than the square of any other number, using a minimum amount of calculation?

Do you see the relationship between this problem and the previous one? Does it help to contextualize in terms of people’s ages?