Math Mistakes that Make the News: Optimization

Good morning! It’s been a while since we had a Math Mistake featured here, but thanks to an article sent my way (thanks MM!), we have some breaking news. Well, “breaking” might be a stretch – the error happened in 1945 and was revealed in 1989, so it’s been a while. But still, it’s an error I hadn’t heard of before and it involves *Calculus*, and specifically checking if your critical point is a max or a min.

So what happened? In 1945 William R. Sears and Irving L. Ashkenas, who worked for the Northrup Corporation, wrote a report comparing the range of a traditional bomber to one that didn’t have a tail and was essentially all-wing (looking a bit like a boomerang). As part of the justification they used CALCULUS to find the shape that gave the best range: set up a function, take the derivative, and see what the critical points are. In this case, there were two: one where almost all the weight was in the wings, and one where it wasn’t. They stated that the all-wing design gave the maximum range.

The problem? That was actually the minimum. A test like the second derivative test can help with that – if the second derivative is negative, then the critical point is a maximum, like ⋂. But if the second derivative is positive, it’s a minimum, like ⋃. 

This mistake was caught within a few years, when Engineer Joseph V. Foa checked the math and told the authors. They admitted it, but also said it didn’t really change things because that was only part of the story. Foa thought that this did too change things, and if this were a couple hundred years earlier there might have been a duel. Instead, the army canceled an all-wing aircraft they’d been working on, supposedly for budget reasons.

So that’s that! Or maybe not – sometimes endings aren’t so clear. Wayne Biddle’s 1989 article below refers to a new B-2 bomber by Northrup that uses the same all-wing design. 

Main source: “Skeleton Alleged in the Stealth Bomber’s Closet” by Wayne Biddle, Science, New Series, Vol. 244, No. 4905 (May 12, 1989), pp. 650-651. (There was a follow-up: “B-2 comes up short” by Wayne Biddle, Science, Vol. 246, Issue 4928 (October 20, 1989), p. 322.)

Monday Morning Math: ChatGPT “research”

Hello everyone!  Apologies for skipping a week – Naz’s Spring Break was a week ago, and I’m still catching up after accompanying a group of students to Hungary. Fortunately, the topic for this week’s Monday Morning Math fell into my lap (well, inbox – thanks Mark!) Last Monday (March 20), the BBC posted posted “The numbers that are too big to imagine”  about infinity.  Here’s a quote from the original article.

Some infinities, [Cantor] showed, are bigger than others.

How so? One of the simplest ways to understand why is to imagine the set of all the even numbers. This would be infinite, right? But it must be smaller than the set of all whole numbers, because it does not contain the odd numbers. Cantor proved that when you compare such sets, they contain numbers that do not match up, therefore there must be multiple sizes of infinity.

If you’ve taken an introduction to Proof class, you might remember the idea of sizes of infinity, but you might also remember that the whole numbers and even numbers are actually the same size of infinity.  The Cantor proof referenced in the paragraph above doesn’t exist.

The reason for the error?  Maybe Richard Fisher, the author, just made a mistake.  Or maybe the mistake was deeper than that.  At the end of the article was the quote:

The author used ChatGPT to research trusted sources and calculate parts of this story.

The error was found quickly, and corrected within a day.  The article now reads:

Some infinities, [Cantor] showed, are bigger than others.

How so? To understand why, consider the numbers as ‘sets’. If you were to compare all natural numbers (1, 2, 3, 4, and so on) in one set, and all the even numbers in another set, then every natural number could in principle be paired with a corresponding even number. This pairing suggests the two sets – both infinite – are the same size. They are ‘countably infinite’.

However, Cantor showed that you can’t do the same with the natural numbers and the ‘real’ numbers – the continuum of numbers with decimal places between 1, 2, 3, 4 (0.123, 0.1234, 0.12345 and so on.)

If you attempted to pair up the numbers within each set, you could always find a real number that does not match up with a natural number. Real numbers are uncountably infinite. Therefore, there must be multiple sizes of infinity.

Yay!  A nice explanation of the different sizes of infinity.  The explanation about ChatGPT also states, “For the sake of clarity, BBC Future does not use generative AI as a primary source or to replace the journalism needed for our articles.“

Indeed – as useful as Artificial Intelligence can be, it doesn’t replace the need for understanding and evaluating what it generates.

Now on sale!

It’s the end of the semester, and I’m pretty much out of markers.  So it’s time to order up some new ones.  And hooray, they are on sale!

 

Well sort of.

 

 

 

Or maybe I’ll wait until the sale ends….

 

A Math Mistake in the News, decimal version

Last month there was a story on BBC.com entitled “Spain’s new submarine ‘too big for its dock'” (https://www.bbc.com/news/world-europe-44871788)

The main part of the story is that Spain’s new non-nuclear submarines were built too large for their docks.  (Hmmm.  Guess that was obvious just from the headline.)

The reason the submarines were too large is that they were redesigned to be bigger than originally planned.

The reason they were designed to be bigger than originally planned is that they were heavier than expected, and so the buoyancy was off, which for submarines is pretty important.  By the time that was discovered it was easier to increase the buoyancy by increasing the volume than by decreasing the weight.

And finally:  The reason that they were so heavy is that someone put a decimal point in the wrong place. According to the article “Navantia gets US help to fix overweight sub” by T. Kington (from http://www.defensenews.com in June 2013, but apparently unavailable now), the former director of the Office of Strategic Assessment at Spain’s Defense Ministry, said “I have been told it was a simple matter of someone writing in one zero when they should have written three.”  I put that in bold, because that small mistake, just twice zero, has taken years and millions billions of euros to (still not fully) rectify.  The contracts for four subs were signed in 2004, the first of the subs was nearly done in 2012 when the mistake was discovered, and now it looks like the subs are all dressed up with nowhere to go.  Poor subs – we look forward to a mathematically successful end to this story.

 

The submarine photo isn’t actually an S-80:  it’s a public domain photo of the USS Chicago (U.S. Navy Photo by Photographer’s Mate 1st Class Kevin H. Tierney. Edited by ed g2s). 

The rate of change of gasoline

Is it better to fill up a gas take once a week for $80, or put in a quarter tank 4 times a week for $20 each time?  That question does have two reasonable answers, depending in no small part on whether you have access to $80 or just to $20 at a time, but what isn’t in doubt is that four quarter-fill-ups at $20 each isn’t actually cheaper overall than one full-fill-up at $80.  Or at least, that shouldn’t be in doubt.

There’s an article about it here, but it doesn’t lessen the confusion at all.

(The reference to the question of  which is closer, the West Cost or the Moon, is a reference to a discussion from a year ago.)

Math Mistake (sort of) – the problem with negatives

It has been a while since we’ve seen a math mistake in the news, but a recent search turned up an old one that I’d never seen (Thanks TwoPi for pointing it out!)  And the funny thing is, it’s not actually a mistake at all – the math is correct.  And that’s the problem.

Back in November of 2007, the National Lottery in the United Kingdom had a new scratchoff ticket for their “Cool Cash” Lotto.  The idea behind the game was the a person would scratch to reveal a specific temperature — say, 15º — and would then scratch to reveal  three more temperatures.  If any of these three numbers was lower than the Chosen Special one (15º in this example), then the person won a prize.  Hooray!

But this was in the UK, which uses Celsius, and negative temperatures are pretty common in the winter.  So the target temperature might be something like –7º, and the three additional temperatures might be –6º, –5º, and –4º.  From a mathematically correct point of view,  that’s not a winning ticket because all the numbers are above –7º.  But people who focused on the numberals 6, 5, and 4, all of which are less than 7, thought they’d won.

It took but a day for this to become a problem, and after no small amount of confusion on the part of customers and shopkeepers, the tickets were pulled.  They had lasted less than a week.  Lottery we hardly knew ye.

For more details, including a video, see the article in the Manchester Evening News:
https://www.manchestereveningnews.co.uk/news/greater-manchester-news/cool-cash-card-confusion-1009701

Another Math Mistake:

This mistake was printed almost a year ago, but it’s still relevant, and math mistakes never go out of style.  This was posted by Richard Fuhr, who I believe is the original author.

The author was looking at an article about the Gobi desert in China, which read in part: “Temperatures may vary up to 95°F (35°C) in one day in the Gobi.”  It also indicated that the average temperature in winter was -40°F (-40°C) and in the summer could be 122°F (50°C)

The -40°F being equal to -40°C is correct – it’s the only place the two temps have equal numerical designation, and I am a little sad that I’ve never gotten to experience it except in windchill form.  The 122°F being equal to 50°C is also correct, and something I have exactly no desire to experience, although it’s still lower than the 129.2°F (54°C) recorded in Kuwait last month.  Both of those conversations can be found by using one of the formulas

• Temp in °C = (5/9) (Temp in °F – 32)
• Temp in °F = (9/5) (Temp in °C) + 32.

The issue is that these are temperature readings, not changes in temperature.  For a change in temperature, the 32 in either formula will disappear, leaving

• Δ°C = (5/9) (Δ°F )
• Δ°F = (9/5) (Δ°C)

This means that a variation of temperature of 95°F would actually correspond to a change of about 52.8°C, not 35°C.  And a variation of 35°C would be a change of “only” 63°F, not 95°F.    It’s not possible to tell mathematically whether the correct variation was  95°F (53°C) or 63°F (35°C), but looking through The Internet at temperature variations, it appears to me that although either one is possible, the printed variation was likely intended to be 35°C, not  95°F.

The photo above is by Doron, with a Creative Commons license.  Thanks to YG for bringing the original article to my attention!

 

How many grams in an ounce?

Converting between units can be hard, as seen before (and before and before).  Fortunately, food containers often include both English units and Metric units.  Unfortunately, those two don’t always match.  Take, for example, Producers Sour Cream.  Their 32 oz container says it has 907 grams, which is about what you’d expect.  The 16-ounce container has half has many.  Not half of 907, but half of that again:  in bold defiance of the laws of physics, it sports a mere 226 grams.

This mistake has apparently gone on for years.  What’s equally strange is that the various nutrition sites that include information about this product also say 16 oz (226g) without comment.     Because, as stated above, units are hard.

Thanks to Philip Bailey for bringing this to our attention!  And speaking of Math Mistakes, as I was, several of the mistakes listed in this very blog are published in the PRIMUS article “Math Mistakes that Make the News” by Yours Truly, which can be downloaded for free during the month of March (2015).

 

Math Mistake…tell your (two) friends

Back in the 80s, there was a commercial for Faberge Organic Shampoo.  And even if the shampoo doesn’t sound familiar, you might have heard of the ad (“…and they tell two friends…”)

Hey, it’s exponential functions!  1 friend tells 2 friends, those 2 friends tell 4 friends, those 4 friends tell 9 friends, those…wait, 9?  Where did that come from?  And then those 9(?) friends tell 16 people.  So it almost works, except that after the photo of 2 people they decided to switch to perfect squares.

Fortunately, a later ad brings the whole thing to a halt before reaching 9:

Good job Faberge people – you skipped the 9!  Of course, this one went straight from 1, 2, 4 to 16 before diving headlong into a grid of 24 people, so I’m not sure it was much of a mathematical improvement.

Threesixty360…your source for commenting on 30 year old math mistakes that have already been well documented.

Decimal Mistake in the News

Decimal points are small, and so easy to lose.  And it appears that many of them were lost on FAFSA (Free Application for Federal Student Aid) forms, which is NOT a place that you would want incorrect data.  According to an official document from July 18, people filling out the form were supposed to round monetary values to the nearest dollar, rather than using exact dollar-and-cents amounts.  But some people put down cents anyway, and the computer didn’t alert them, or tell them there was a problem.  No, it slyly accepted the amounts, and then threw all the decimal points in the trash, so [as the official memo said], and income that had been recorded as $5000.19 was suddenly interpreted as $500019, which is one heck of a sweet income and probably enough to disqualify you from most financial aid.

This didn’t happen with just a couple people, either – The Wire says that 200,000 people are likely to be affected.  And because it’s more than just a couple, schools have to look at all those applications, every single one, to catch any errors.  Those errors might be that people didn’t get aid who should, which is a bummer, but it could also mean that people got too much money.  That doesn’t sound as bad initially, but the July 18 memo says, ” If such aid has already been disbursed the institution may need to change awards and return (or have the student return) any overawarded funds.”  I can’t imagine that it will go over all that well for a school to tell someone to give back money that was promised, so I suspect this messiness will last a while.

Hat tip to Yousuf for pointing out this article!

Negative Zero

Zero degree (Celsius) is cold.  But you know what’s really cold?  Negative zero.  At least according to the sign that our colleague Nicole saw in Canada.

 

Math Mistakes and Misplaced Measuring

In an unfortunately tribute to Pi Day and the importance of mathematics, there was an article in the New York Times yesterday (March 13, 2012) illustrating that the people who need to measure parts don’t always know how:

“The employee responsible for finding a replacement part for a tower crane that ultimately collapsed on the Upper East Side in 2008, killing two workers, testified on Tuesday about his own difficulty with the basic math of measuring key components. Tibor Varganyi, whose formal education ended in the ninth grade in Hungary, struggled how to measure the distance between the roughly 30 bolt holes around a piece of the turntable assembly. He decided to use a ruler.”

The article (“Worker Tells Court He Lacked Math to Measure Crane Part” by Russ Buettner)goes on to explain how the measurements didn’t match up with expectations, so he switched to a protractor, which also didn’t work.  This particular replacement part was never used, and the article is primarily about the prosecution’s argument that the company wasn’t worried about the lack of expertise or safety, instead focusing on profits, but the description is still worrisome.

That’s depressing.  We’d better recover by looking back at some old Pi Day Sudokus.

Granola Fail

From a recent Nature Valley ad in the London Metro newspaper:

Perhaps the second bar is twice as delicious as the first.

Via Language Log.  Photo from Spiderham.

Math Neglect on Glee

The folk from Glee paid unintended homage to the title of this week’s episode (“A Night of Neglect”) by showing Mr. Schuester forgetting his basic math skills.  Actually that’s not entirely true; he does math in his head correctly as he explains his plan to use salt-water taffy to earn money to go to Nationals in New York:

When I was a student here we paid for our entire trip to Nationals selling this….  So, to make $5000 at 25 cents apiece, we need to sell 20,000 pieces of taffy.

So far, so good.  But wait, what’s that equation in the background?

Poor Will…he didn’t even notice that the equation wasn’t quite right (and neither did the four members of the Academic Decathalon team).  But don’t worry, we understand how busy this time of year is, what with all the projects and end of the year assignments coming due.  So shall we just fix that up for you?

There, all better.   Now you can go concentrate on raising that money.  Just be sure to have someone else in charge of the ledger.

Rounding Up – Way Up

Ever heard of Dudeney numbers?  Neither had I, until yesterday, when I discovered them completely by accident while reading (Wikipedia, what else?) about narcissistic numbers.  A Dudeney number (named after famous English mathematician and puzzle author Henry Dudeney) is a number that is the cube of the sum of its digits.  For example,

There are only six Dudeney numbers.  Neat numbers, but I was a little disappointed by that.  What to do next?

Generalize, of course!  Generalized Dudeney numbers (discussed here, but the link appears to be dead, so I used Google’s cached version) are numbers that are some power of the sum of their digits:

The largest number on the above site is , which has 147253 digits.  The site links to Wolfram Alpha to confirm this.  Here’s where it gets weird:

How many digits is that?  About ?  About a million?  What kind of rounding is that?  It gets worse.  Try a number with just 100,002 digits (despite what Alpha says).  I think Alpha is a great tool, and I’ve had (far too much) fun playing with it, but I’m a tad disappointed (that’s twice in one post).  So, hey, get on that, Wolfram.