Puzzle 1: Some puzzles have a charm because of the strange questions they ask. One puzzle tells you to suppose you are the driver of a double decker bus in London, and then proceeds to tell you how many passengers get on and off at various stops, so that you are distracted into keeping track of the sum, and then you are asked "So what was the bus driver's name?". Of course there's an answer, but you can't even imagine answering the puzzle because the question seems absurdly unrelated to the data.
In this puzzle, similarly, you don't seem to be given any information that could tell what the actual formula is, and yet a very specific result is requested. One's solving schemes naturally begin churning out guesses about, perhaps, some special property of the number 129, say, that will make it possible to determine the result of the formula (though not the full formula) in this special case. But this is all trickery designed to keep you from the answer!
As I was thinking about this puzzle, a friend looked at it and said, "Well, this is silly! What's to stop the function from being the identity function?" I looked for a minute, and realized that she had made an important statement:
The identity function f(n)=n is a solution.Having worked on puzzles for a while, a new thought immediately comes to mind:
Is the identity function perhaps the only solution?
But now the problem begins to seem like one we can handle. It was easy to see that, for instance, the fact that the function is "onto" guarantees that there must be some number m so that f(m)=0. Could this number be anything other than 0? If m is not 0, then it must have the form m=p+1. Therefore, it would be true that:
f(f(p)) < f(p+1) = f(m) = 0
but that can't happen because we know that the lowest value of f
is 0. So therefore, we know that f(0)=0 and 0 is the only argument
whose functional value is 0.
Now you can see where this is leading. A similar argument shows you that there must be some number m so that f(m)=1, but that this number cannot be anything other than 1, and so on. Hence, the function f must be the identity function, and hence, "trivially", f(129)=129.
Back to the The Function Puzzle.