How Many Half-Lives Do You Have?

Do protons live forever or do they decay with a half-life of around 16 billion trillion trillion years? That’s an eternity considering the universe is thought to be less than 14 billion years old. Yet, as Natalie Wolchover recently described, the fate of physicists’ beloved grand unification theories — the idea that the forces of nature were unified at the beginning of time — rests on finding that protons are in fact mortal after a humongous half-life. It’s a question scientists have yet to settle.

I find the half-life concept fascinating. It forces us to think in unintuitive ways. Let’s start with two simple questions that get to the essence of what a half-life is. You have to answer both questions in just one minute (consider half a minute to be each question’s half-life).

OK, start your half-life timer. Question 1:

In a bucket filled with nutrients is a bunch of amoebas. Every day, their population doubles. If it takes 48 days for the bucket to be filled with amoebas, how long did it take for the bucket to become half full?

Question 2 is based on the title of this puzzle:

How many half-lives does a pound of radioactive material have?

If you’ve never seen these types of questions and you stuck to the time limit, you may have jumped to the wrong conclusion.

The first question is very similar to one of three problems that constitute the Cognitive Reflection Test, originated by Shane Frederick, a decision theorist now at the Yale School of Management. According to Frederick, there are two general types of cognitive activity, the first of which is carried out automatically, without reflection, while the other requires conscious thought and effort.  Each of his three problems has an obvious or impulsive response, which originates from the automatic process, but is incorrect. In order to activate the second process, you must recognize that your first answer is incorrect, which requires you to reflect on your cognition. I remember encountering a question like the first one as a 7- or 8-year-old proud of my arithmetical skills. I fell for the bait hook, line and sinker, utterly confident of my answer, which turned out to be the obvious, incorrect one. I’ve been trying to reflect ever since.

In order to intuitively understand and make sense of half-life, like the Cognitive Reflection Test, you have to reject the first impulsive answer and try to consciously reach the right way of thinking, eventually making it part of your intuition or “mental heuristics” toolbox.  It’s well worth the effort. If you need to peek, drag your mouse to select the text following “Answer 1” and “Answer 2” below.

Answer 1: 47

Answer 2: Theoretically, infinite. In practice, a very, very large number (we’ll discuss more precise answers in the solution column).

To think naturally about half-life, first, remember that it arises wherever there is exponential decay, or, as with the amoebas, exponential growth. We are intuitively more accustomed to linear change. Second, there is nothing magical or sacrosanct about decreasing by half — we use this fraction only for convenience. The concept and the mathematics work just as well for any other fraction, such as a third or a fifth. A more natural fraction, used in many mathematical and physical applications (such as the time constant in resistor-capacitor circuits), is 1/e, where e is the base of natural logarithms, 2.71828…. (1/e is about 0.368, or 36.8 percent). A final important concept is that the behavior that gives rise to half-life in a group is actually completely dependent on the properties of the individual. When a bunch of radioactive atoms decrease to half their number after one half-life is over, each atom knows nothing about any other atom, or the size of the group it belongs to. The group’s magical reduction to 50 percent every time one half-life passes emerges simply out of each atom’s exact same probability of decaying within the given time, independent of all the others.

The fact that particle decay is probabilistic means that, as in the case of proton decay, you just have to collect a large bunch of them and hope that some of them disintegrate within the time you have at your disposal. If the particles don’t oblige, there’s nothing you can do. That’s why one scientist compared the lack of results in the proton decay to waiting for your spouse to come home: “If they’re 10 minutes late, there’s simple explanations for that. An hour late, maybe those explanations become a little less plausible. If they’re eight hours late … you begin to worry that maybe your husband or wife is dead. So the point is, at what point do you say your theory is dead?”

While we have no grand unified theory to grind, here are a couple of questions to get the flavor of half-life calculations for those who need real numbers to sink their teeth into.

Question 3:

Let us say you manage to obtain 5 atoms of the radioactive isotope of unobtainium, of Avatar fame. After exactly one year, 2 atoms have decayed. You want to figure out the half-life of the substance, and like a true scientist you seek a range that has a 95 percent probability of containing the true value. What is your range for the plausible half-life of unobtainium?

Question 4:

In the above scenario, for your second year, you obtain 30 atoms of the substance. You need to divide it into three portions — A, B and C — according to the following rules. When you inspect the portions after exactly one more year, you would like A to have exactly 6 undecayed atoms, B to have 7 or 8, and C to have 4 or 5. Remember, you don’t know the half-life exactly. How many atoms must be in A, B and C initially in order to maximize your chances of getting the precise results you want?

That’s all for now. Happy New Year, and happy puzzling! The solution column will arrive with a half-life of 15 days.

Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt. And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION” (it will not appear online, so solutions to the puzzle above should be submitted separately).

Note that we may hold comments for the first day or two to allow for independent contributions by readers.

Update: The solution has been published here.