How to Measure Time Without a Stopwatch

How can you measure time without using a stopwatch? You could use the movement of the Sun across the sky, watch a pendulum swing, or burn some very special string. Keep on reading to find out how it works!

Jason Marshall, PhD,
January 14, 2017
Episode #250

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Burning FuseA math puzzle a day keeps your brain saying "Yay!"

I know that’s not the most memorable saying in the world, but it’s definitely true—puzzles are a fantastic workout for your brain. As such, you’d be wise to try your hand at tackling at least a few different kinds of puzzles every week. And the best part of this is that mental exercise like this is fun!

To help you in your endeavor to start puzzling more, today we’re going to take a look at a great brain teaser that I recently ran across. This puzzle is all about time and how you can measure intervals of time in a rather unusual way: by burning bits of specially crafted string. How does it work? And what’s the big brain teaser?

That’s exactly what we’ll be talking about today.

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!


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