# Maths on the Back of an Envelope

How powerful was the first atomic blast? How many cats are there in the world? How much is China's coronavirus epidemic impacting its economy?

These are three very unrelated questions. If you were tasked with responding to them, how could you possibly come up with reasonable answers to any of the three?

Each of these questions are examples of what might be called "Fermi problems", which are named after Italian-American physicist Enrico Fermi. In addition to his pioneering work in nuclear physics, Fermi was also known for doing back-of-the-envelope calculations to come up with estimates that made real world sense even when nobody had any idea of what the right answers might be.

In Rob Eastaway's latest book, Maths on the Back of an Envelope (more affordably available in the UK), which discusses 'clever ways to (roughly) calculate anything', he tells the story of how Fermi estimated the force generated by the first-ever atomic bomb blast using confetti and a few quick calculations just after it happened:

The story goes that Fermi and others were sheltering from the explosion in a bunker about six miles from ground zero. When the bomb went off, Fermi waited until the wind from the explosion reached the bunker. He stood up and released some confetti from his hand, and when it had landed, he paced out how far the confetti had traveled. He then used that information to make an estimate of the strength of the explosion. We don't know for certain how Fermi did this, but it probably involved him estimating the wind speed and working out how much energy was required to push out a 'hemisphere' of air from the centre of the explosion.

Fermi's estimate of the bomb's strength was 10 kilotons. Later, more rigorous calculations revealed that the real strength had been nearer to 18 kilotons, in other words Fermi's answer was out by a factor of nearly two. Anyone submitting an answer that far out in a maths exam would probably get no marks, yet Fermi got huge credit for the accuracy of his back-of-envelope answer. The important thing was that his answer was in the right order of magnitude, and gave scientists a much better understanding of the potential impact of the weapon they were now dealing with.

At the time, the power of an atomic blast was such an unknown that other scientists working on the project had been concerned enough before the test to do some pretty sophisticated math just to provide the reassurance to themselves that the atomic bomb they built would be unlikely to set off a chain reaction that would ignite the Earth's atmosphere and destroy the planet. Getting the order of magnitude of the actual event correct under such uncertainty is what makes Fermi's quick math such a big deal.

So what does all that have to do with the other two questions?

It has everything to do with how you can take information you know and logically chain it together with reasonable guesswork so that you can use it to extrapolate an estimate for the answer you're trying to reach. In his book, Rob Eastaway describes how he once estimated the global population of cats while speaking to a school assembly (we've added the image as an illustration because, well, cats):

Let's assume that most cats are domestic.

Some people have more than one cat, but usually, a household has only one cat, if any at all.

In the UK, and thinking of my own street as an example, it seems reasonable to suppose that there might be one cat in every five households.

And, if a household contains on average two people, that means there is one cat for every 10 people.

So, with 70 million people in the UK, let's say that there are, perhaps, seven million cats in the UK.

So far, so good. But what about the number of cats in the rest of the world? It seems unlikely that cats are as popular in countries like India or China as they are in the UK (although what would I know? Remember, this is purely guesswork on my part), therefore, I'd expect the ratio across the world to be smaller than it is in the UK - maybe one cat for every 20 people?

So, with eight billion people in the world, that suggests there are maybe:

8 billion ÷ 20 = 400 million cats

It doesn't seem an outrageous number.

A member of Eastaway's audience searched the question on Google and it returned 600 million as the answer, which Eastaway takes as a sign that the math he did was on the right track, or rather, that whoever came up with that larger estimate likely went about working up their estimate using similar methods.

And that brings us to China, which is where economist David Tufte recently did some back-of-the-envelope math to assess how hard China's economy has been hit by the COVID-19 coronavirus epidemic that has killed thousands and sickened tens of thousands in the nation.

This may be the best real time estimate yet on what COVID-19 has done to the Chinese economy. China’s power plants run mostly on coal. China’s coal consumption appears to be down between 20 and 45%.

This is measured in days since the Chinese New Year, which fell on January 25 this year. So, they’re usually down for about 10 days after that, and this slowdown has stretched on for almost a month now.

To get that to GDP we need to know China’s energy elasticity. A plausible value for any country is around one, estimates from 15 years ago suggest 1.5 is more suitable for China. Here’s the back of the envelope calculation:

• Choose a round number for China’s GDP like \$20,000B/yr
• Coal consumption is down 20% to 45%
• The elasticity suggests a hit of 30% to 70% for GDP
• That’s \$6,000B/yr to \$14,000B/yr if it’s a discrete jump. It isn’t, so looking at that typical slope showing recovery by about day 25 in most years, that slope suggests effects so far that are perhaps half of that as China built up to a sustained shortfall.
• This shortfall is new and gradual, let’s say it’s about 1/25th of a year so far (about 2 weeks). That converts to a GDP loss of between \$120B and \$280B so far, or –0.6% to –1.4% of annual GDP in total.
• China’s economy in 2020 is roughly the size of the U.S. economy in 2008-9. During the worst parts of that recession, the U.S. economy was off \$20B in 2008 III, \$85B in 2008 IV, and \$45B in 2009 I.

All of these numbers are sketchy, but the suggest that the effect of COVID-19 on China over a few weeks is already comparable to what a large recession did to the U.S. in a few quarters.

Given China's role in global supply chains, both as producers and as consumers, that magnitude of economic impact will affect other national economies, bringing a global recession into view, which is contributing to why the world's stock markets are plunging.

It will be a long time before all the actual damage has been tallied, but until then, the kind of estimations we can do using back-of-the-envelope maths will give us the best indication of how things are going. It's also why we can recommend Maths on the Back of an Envelope as entertaining reading material that you can actually put to use to answer questions that, on first glance, may appear to be unanswerable.

### Previously on Political Calculations

Image credit: Jari Hytönen

# Black Friday Sale Calculator

Black Friday is another uniquely American tradition that arose as many Americans have the day off work following the national Thanksgiving holiday, which about a month before Christmas, provides an opportunity to shop for gifts for that upcoming holiday.

With so many shoppers hitting their stores, many retailers looking to claim a larger share of all the cash that will be spent on Black Friday will offer special discounts to their customers, which raises an obvious question. Are their special sale prices worth the hassle of shopping at their stores?

Hanna Pamula built a tool to find out, which Lifehacker profiled earlier this month:

It certainly feels like everything is on sale all the time these days. But not all sales are created equal. It’s especially important to recognize this during the holidays, when shopping holidays feature complex discount schemes that sometimes carry equally complex limitations.

If mental math isn’t your strong suit, check out the Black Friday Calculator. “A lot of these seemingly genuine sales are often math traps,” said Hanna Pamula, a Ph.D. student in Poland who built it for The Omni Calculator Project. She includes nine different discount scenarios —percent off, buy-one-get-one, and a discount for buying multiple items, to name a few—to help you determine whether a discount is worth your time and money.

We took the tool for a test drive, and so can you, since we've embedded it below. If you're accessing this article on a site that republishes our RSS news feed, please click through to our site to access a working version.

Black Friday Calculator

Overall, we would describe the tool as a fun personal finance application. Since it covers a number of different kinds of promotions that retailers offer, from the simple "X% off" sale to "Buy one, get Y% off a second one" to more complicated ("Get an additional W% off our Z%-off sale price") promotions, it will be useful in evaluating many of kinds of discounts you will inevitably encounter while shopping.

On that latter count, it can help you compare competing sale prices, letting you know which retailer's promotion delivers the biggest discount.

One thing we found was a bit of a hassle is the default setting for sales tax, which we would argue should be set to "No" rather than "Yes" for American shoppers, since most sale prices in the U.S. do not roll in the multiple levels of sales taxes that apply in the various jurisdictions that impose them across the nation. That's less a concern for other countries that impose national value-added taxes on sale transactions, but for Americans, it will add a couple of steps between data entry and results.

The Omni Calculator project is a site that features several hundred calculators with tools for doing math that applies across a wide variety of fields and applications. We've been in the online tool building business for a long time, where we're always happy to find new resources that online applications easier. If things slow down enough for you this Black Friday weekend, do check out their other tools!

Image credit: Artem Beliaikin

# Jack Bogle on Investing

John Bogle, the man who made passive, low-cost index investing a real world thing and who, as a result, built Vanguard into one of the world's largest investment management firms, passed away on 16 January 2019 at Age 89.

The idea of index investing that Jack Bogle championed proved to be very a big deal, which is why the index fund made Tim Harford's list of 50 Inventions That Made the Modern Economy (the UK edition is 50 Things That Made the Modern Economy), where we can strongly recommend the 10-minute podcast episode of the related BBC radio series if you want to learn more about its history.

If your available time is shorter than that, Jack Bogle once claimed that he could teach the essentials about investings in just a few minutes. In 2010, the Bogleheads' Ricardo Guerra put him to the challenge, where he distilled a lifetime of learning about successful investing into 3 minutes and 42 seconds....

Back in October 2006, we participated in a chapter-by-chapter review of the original edition of The Bogleheads' Guide to Investing (now in its second edition), where we had the honor of reviewing the final chapter of a book that sought to capture Jack Bogle's wisdom....

That was a lucky break for us, because the final chapter summarized all the lessons presented throughout the book, which gave us the opportunity to further condense Jack Bogle's thoughts on investing into just six lines, although with quite a few links to follow for deeper insights gleaned by the other participants in the project:

Today, millions of people are considerably richer than they might otherwise have been because of what Jack Bogle wrought. That's one hell of a legacy in the financial world!

# Jack Bogle on Investing

John Bogle, the man who made passive, low-cost index investing a real world thing and who, as a result, built Vanguard into one of the world's largest investment management firms, passed away on 16 January 2019 at Age 89.

The idea of index investing that Jack Bogle championed proved to be very a big deal, which is why the index fund made Tim Harford's list of 50 Inventions That Made the Modern Economy (the UK edition is 50 Things That Made the Modern Economy), where we can strongly recommend the 10-minute podcast episode of the related BBC radio series if you want to learn more about its history.

If your available time is shorter than that, Jack Bogle once claimed that he could teach the essentials about investings in just a few minutes. In 2010, the Bogleheads' Ricardo Guerra put him to the challenge, where he distilled a lifetime of learning about successful investing into 3 minutes and 42 seconds....

Back in October 2006, we participated in a chapter-by-chapter review of the original edition of The Bogleheads' Guide to Investing (now in its second edition), where we had the honor of reviewing the final chapter of a book that sought to capture Jack Bogle's wisdom....

That was a lucky break for us, because the final chapter summarized all the lessons presented throughout the book, which gave us the opportunity to further condense Jack Bogle's thoughts on investing into just six lines, although with quite a few links to follow for deeper insights gleaned by the other participants in the project:

Today, millions of people are considerably richer than they might otherwise have been because of what Jack Bogle wrought. That's one hell of a legacy in the financial world!

# Reversing 300 Million Years of Continental Drift

If you're a fan of the theory of dynamic plate tectonics (and really, who isn't?), you can spend a lot of time checking what that has meant for where you live by playing with Dinosaur Pictures' Ancient Earth Globe app, where you can plug in your address and see how the Earth where you're at has changed over tens and hundreds of millions of years!

We did that, where we plugged in a famous address in Washington D.C., which centered the globe in today's world on the Western Hemisphere. We then moved backwards in time, until we got to a point where the spot where Washington D.C. would eventually be located moved into the opposite hemisphere. The following animated image shows 300 million years worth of that continental drift in reverse....

We set up the animation so that each of the frames would be displayed for eight seconds, which should allow enough time to read some of the additional information that the app presents.

Do check it out - although we stopped at 300 million years, the app goes back some 750 million years.