Two weeks ago I posted a puzzle sent to me by my nephew Matthew:

1 is 3, 3 is 5, 5 is 4, and 4 is cosmic.

Why is 4 cosmic? What happens to the other numbers?

As I mentioned in the earlier post, I slept on this before solving it. I initially thought about the riddle “mathematically,” but that was no help.

A mathematical way to think about the phrases involving the word “is” in the statement of the problem is to consider them as mappings. That is, consider a function (or mapping) $f$ that maps 1 to 3, 3 to 5, and so on: $f(1) = 3$, $f(3) = 5$, $f(5) = 4$. I guessed that “$4$ is cosmic” meant that 4 is a fixed point of this mapping; that is, $4$ is mapped to itself so $f(4) = 4$.

So the question is, can we guess a formula for $f$? What came to me in a flash is that it is possible to express the action of $f$ in words: $f$(number) = number of letters in the English word for the number.

Thus, $f(1)$ = number of letters in “one” = $3$; $f(2)$ = number of letters in “two” = $3$; and so on. And $4$ is indeed a fixed point of this action because the word “four” has $4$ letters.

Now, it could be that “cosmic” means that whichever number you begin with, by repeatedly applying the function $f$ you will end up at $4$. To check this, we would first have to determine if there are any other fixed points of this action. (I assume the domain of the function is the whole numbers.) There are no other fixed points, but I will let you check this.

The other thing that must be checked is that there are no “cycles” in this action. For example, perhaps $11$ gets mapped to $8$, which gets mapped to $7$, which gets mapped to $11$, and round and round we go indefinitely. This does not happen either, but again I’ll let you check this for yourself.

In checking each of these facts, we are helped by knowing that the “word-lengths” of numbers increase much more slowly than the numbers themselves. Numbers less than $1000$, for instance, are mapped to numbers in the twenties or less.

Functions that are applied repeatedly amount to iterative procedures, and have many applications. For example, Picard’s theorem is an iterative procedure for solving certain ordinary differential equations. An important method for solving equations of many types is to cast the equation into the form $f(x) = x$; if the function $f$ has a fixed point and satisfies other properties (which depend on the type of equation being considered), then a fixed-point theorem (that guarantees a solution exists) may be available. The solution process then involves an iterative procedure in which the solution is approximated more and more accurately, with the limit being the fixed point. A general overview of fixed-point theorems is here.

All kinds of behaviours are possible with maps that are applied iteratively. For example, fractals are generated in this way. Certain dynamical systems are defined via iterative maps; the logistic map is a famous example, and one that provides an interesting entry into this field, as it includes both deterministic and chaotic behaviour, and interesting jumps in behaviour from one type to another depending on the value of a parameter.

One of the most famous unsolved problems involving an iterated map is the Collatz conjecture (also known as the hailstone conjecture), which can be phrased in the form of a game. The game works like this: start with any natural number. If the number is even, then divide it by $2$. If the number is odd, then multiply it by $3$ and add $1$. Repeat the process with the resulting number. The conjecture is that if you repeat the process continually, you will eventually end up with the number $1$, no matter which natural number you begin with.

Nobody has yet been able to prove that the Collatz conjecture is true or false, but you might have fun exploring it.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 21 January 2021.)