How to Measure Time Without a Stopwatch
How Can You Measure 45 Minutes?
Imagine you’ve been given several pieces of string with varying lengths and thicknesses. Not only do the lengths and widths of the pieces vary, each piece of string isn’t even uniform in width along its own length. In other words, they get thicker and thinner (by different amounts and in different places) as you go from one end to the other. While all of the pieces are therefore different, they have one thing in common: if you light one end on fire, it will always take exactly 1 hour to burn through to the other side. But since each piece gets thicker and thinner as it goes, a given piece of string doesn’t necessarily burn at an even rate. By which I mean that a string doesn’t necessarily burn half its length in 30 minutes—all we can say is that it burns its entire length in exactly 1 hour.
Is it possible to use these pieces of string to measure a 45 minute time interval?
So that’s the setup. Here’s the question: Is it possible to use these pieces of string (as many as you want) to measure a 45 minute time interval? And, if it’s possible, how would you do it? As with every puzzle, it’s a lot more fun if you give it a try before finding out the answer. So I encourage you to pause for a few minutes to give it a go. Then, when you’re ready, continue on for the answer.
A Simpler Problem
Before we solve today’s puzzle, let’s imagine a slightly simpler puzzle in which the pieces of string we’re given all burn at a uniform rate. In this case, you could solve the puzzle simply by folding a single piece of string in half, and then by folding this in half again to make creases in the string at 1/4, 1/2, and 3/4 its total length. Since this string burns at a uniform rate, all you have to do to measure 45 minutes of time is simply light one end of the string on fire and wait until it burns to the mark that’s 3/4 of the way towards the other end. Easy!
But, sadly, the pieces of string we’re given in this puzzle are not so well behaved in that they don’t burn at an even rate (since they get thicker and thinner at various points along their length). Which means we can’t simply fold a piece of string into quarters to count off time. But that doesn’t mean we’re out of luck … because we can do something much more clever!
How to Solve It — Step 1
Instead of lighting just one end of a piece of string on fire, we start by lighting both ends on fire at the same time.
Here’s what we need to do: Instead of lighting just one end of a piece of string on fire, we start by lighting both ends on fire at the same time. What does this do for us? Well, the uneven thickness of a piece of string means that we can’t say how fast the fire will move at any given moment as it’s burning from one end to the other (or the other way around). But, nonetheless, we do know that the two flames burning towards each other must come together and meet (although we’re not quite sure where) after precisely 30 minutes! Make sense?
That means we now have a way to measure 30 minute intervals in addition to 1 hour intervals. Of course, what we really want is to measure a time interval of 45 minutes. So we’re not done yet. What do we do now? Before answering that question, once again if you haven’t figured it out yet, I encourage you to stop for a minute and think about it before continuing on.
How to Solve It — Step 2
OK, so what can we do to measure a 45 minute time interval instead of just 30 or 60 minute time intervals? The trick is that right after we simultaneously light that first string from both ends, we need to light another string from only one end. Then, when the two flames of the first string come together—which we’ve already figured out marks the 30 minute point—we need to light the other end of the now half-burned (in terms of time) second string. At this point, the two flames of the second string will start burning towards each other, and they will meet in the middle after exactly another 15 minutes pass.
Putting all of this together, we see that the burning ends of the second string must meet a total of 30 + 15 or 45 minutes after we started this whole process of setting strings on fire. And thus, by applying a bit of clever thinking, we have managed to measure a time interval of 45 minutes. Pretty cool, right?