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….

12 tables, 24 chairs, and plenty of chalk

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….

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). *

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.)

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*

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!*

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).*

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 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!*

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.

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.*

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.

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.

Here’s a good rule of thumb: if you’re trying to calculate how much money to send an insurance company, it’s probably a good idea to round up. That’s a lesson that La Rosa Carrington learned the hard way.

Carrington had health insurance under her job, and when she lost her job she was allowed to continue her health insurance under federal COBRA law. Trouble is, she didn’t get a bill so she estimated the amount she would have to pay: her payments were “a little over $471.87 per month” (according to *The Gazette* in Colorado Springs, where the story first appeared on July 6) but because of the 2009 American Recovery and Reinvestment Act she only had to pay 35% of that.

Carrington didn’t get a bill from Discovery Benefits, and yet she knew it was important to keep up the payments, **especially** because she was also undergoing chemotherapy for leukemia so details like current health insurance coverage were totally non trivial matters. She sent them a check for $165.15. Trouble is, Discovery Benefits said she owed $165.16, and canceled her coverage. She called, they refused to budge, and finally the supervisor did the calculation herself and decided that rounding the amount to $165.15 was actually right, or at least reasonable, and the penny was paid [either by the company or by a person in the company; it’s not clear which].

So be warned: sending in that extra penny might be good insurance for your insurance.

*The story could end there, since rounding is all mathematical in and of itself, but there’s a tangent that I’m still wondering about: what’s the deal with the monthly payments being “a little over” $471.87 each month? If the annual dues were $5662.46, for example, then the monthly payments would be $471.8716666…, which would round to $471.87, but 35% of the original $471.871666….. would actually be $165.155083333… which does round to $165.16 using conventional models of rounding. It seems plausible to me that the Benefits Computer was just rounding, and not necessarily rounding up all the time, and that the multiple rounds gave a difference of a penny, which would make this a story not about rounding up versus rounding down, but about the compounding of rounding errors. I looked at a few different reports on this, though, and never saw mention of this so it’s possible that the Benefits Computer was automatically rounding up for all rounding as implied.*

Probably more than a few, really.

There have been a few Math Mistakes in the News this summer with regard to insurance companies, though unfortunately I can’t seem to dig up what all of the actual mistakes are.

Back on May 5 the California Department of Insurance released a press statement that Anthem Blue Cross had been making some math mistakes and they were going to be under extra scrutiny. According to the statement:

The errors identified included:

- Error #1: Double counting of aging in the calculation of underlying medical trend for the projection of total lifetime loss ratio.
- Error #2: Anthem overstated the initial medical trend used to project claims for September 2009 for known risk factors.
Both of these errors are errors of math and not differences in actuarial opinion.

I didn’t see anything about this costing a particular amount of money, though a June 25 article from the Los Angeles Times indicates that they canceled a rate increase of up to 39% for many of their California customers customers as a result.

Then, less than two months later, Aetna Inc. also had some math woes. According to the same LA Times article,

Connecticut-based Aetna Inc. had sought an average 19% increase in rates for its 65,000 individual customers, but pulled back after multiple math errors in its paperwork were found by its own staff and by an independent consultant working for the state.

I was tempted to write “Bummer” but there’s really nothing bummerish about not having a 19% rate increase. There’s no direct statement of what the mistakes are, just that “There were multiple errors … in the way [Aetna] annualized premiums and in the compounding of the rate increase,” according to California Insurance Department spokesman Darrel Ng.

Another article that same day on a Consumer Watchdog site, quoted Watchdog president Jamie Court as saying, “It’s amazing how insurers are making mathematical errors when they’re not used to regulators checking their math.”

Incidentally, a similar error was more recently discovered across the pond. According to citywire,

Yorkshire and Clydesdale Bank today said it is in the process of mailing around 18,000 variable rate mortgage customers to apologise for miscalculating their monthly repayments and to suggest ways customers can repay what they owe.

The bank said the calculating error, which was exacerbated by last year’s unprecedented falls in interest rates, led to the bank collecting less than the contractual minimum monthly payment required for customers to pay their mortgage within their agreed term.

Moral? Check your math.

*Seriously, though this topic is serious enough, several of our math grads in recent years have gone into different aspects of insurance and finding errors can be a pretty important part of their jobs, whether it’s part of the official description or not.*