Parade Magazine is a supplement that most newspapers, although not the hoity-toity New York Times, insert into their Sunday edition. Since 1986, nearly every issue has included a brainteaser column by Marilyn vos Savant. Besides her impressive feat of publishing weekly for nearly thirty years, vos Savant may be best known for having once been listed in the Guinness Book of World Records as the person with the highest recorded IQ – an “accomplishment” which, until recently, she trumpeted at the bottom of every “Ask Marilyn”.

Naturally, I read every one hoping to catch her in error. She rarely satisfies this craving but every now and then, like today, she messes up. In fact, when I read her January 25 puzzle, I knew she had presented an almost identical problem before (turns out it was in 1992) and answered it incorrectly. I wondered whether she would tacitly admit her prior error by providing the correct response. She did not.

The question is which is better: A starting salary of $30,000 with (A) an annual $1000 raise or (B) a twice-yearly $300 raise? Now, your immediate response as mine was 23 years ago and this morning is probably (A), right? I mean $1000 more over one year beats two $300 raises during the same period but vos Savant says the answer is (B).

Here’s her explanation:

Say it’s Jan. 1, 2015, and you choose the $1,000 raise at the end of the year. In 2015, you earn $30,000. But if you choose the $300 raise, you earn $15,000 during the first six months and $15,300 during the second six months. In 2015, you earn $30,300. With the $1,000 raise in effect for 2015, you earn $31,000. But wih the $300 raise, you earn $15,600 during the first six months and $15,900 during the second six months. In 2016, you earn $31,500.

Sounds persuasive doesn’t it. The problem is that vos Savant wrongly compresses the distribution of each $300 **annual** raise such that it’s paid in full in half a year. In fact, the years go as follows:

2015: (A) $30,000 (B) $30,150 ($300 raise is in effect for 1/2 the year)

2016: (A) $31,000 (B) $30,750 ($30,000 + 1/2 X $600 + 1/2 X $900)

2017: (A) $32,000 (B) $31,350 ($30,000 + 1/2 X $1,200 + 1/2 X $1,500)

2018: (A) $33,000 (B) $31,950 ($30,000 + 1/2 X $1,800 + 1/2 X $2,100)

After the first year, the gap favoring (A), not vos Savant’s choice (B), becomes wider and wider. She might respond that because the question provides for either a “$1,000 raise every year” or a “$300 raise every six months” the extra $300 each 1/2 year is supposed to be paid in full within the 1/2 year. This explanation fails because both (A) and (B) assume a $30,000 annual salary so any raise would likewise be distributed over the course of the year.

Marilyn would be on less shaky ground if the employee were offered either (A) $30,000 annually to start with a $1,000 raise each year or (B) $15,000 per 6 months with a $300 raise at the end of each period. But that is not how the question is phrased. She might also have an argument if the offers called for bonuses which, by definition, are paid in full when conferred. But again the question clearly defines the employee’s choice as between two different pay raise schedules. Simply put, vos Savant got the same problem wrong twice over a 23 year span.

Why am I devoting time to this? Well, vos Savant reaches an awful lot of Americans each week who have faith in her reasoning despite the fact that she does make elementary mistakes.

Math errors may be trivial but what about her position on global warming which is prominently featured on her home page and reprinted in full below:

I believe our planet is warming, of course. Although I don’t know how much humans have contributed, I would be surprised if it’s close to 100 percent. So, knowing how vastly the Earth’s climate has varied over its four- to five-billion-year history, I doubt that we can succeed in stabilizing it the way it is now. Regardless, we must try, if only to buy time.

But we can do better than that. I’d like to see our government supporting more adaptive, positive responses to the warming instead of assuming that we must concentrate all of our efforts on scaling back and attempting to maintain the world as it is today. There’s no significant downside to encouraging innovative programs designed to maximize every potential advantage of warming and minimize the disadvantages wherever it’s feasible.

The shift in course would be small, yet I believe that we would be heading in a more productive direction over time. Along with mending our ways, we would be in a safer situation overall if we also develop more coping mechanisms for whatever nature alone may be handing us. Call it “technological adaptation.”

At first blush, vos Savant is taking a reasonable down-the-middle approach. At least she doesn’t deny the basic fact that the planet is getting hotter and hotter. But starting with the second sentence, she descends into absurdity: “I don’t know how much humans have contributed but I would be surprised if it’s close to 100%.” Note what vos Savant does here. She refuses to concede that humans have any responsibility for the approximately 1.4 degrees F rise in global temperatures since 1880 with most of that since 1980. She fails to note that NASA, NOAA, and the Intergovernmental Panel on Climate Change, along with nearly every reputable scientist who has studied the issue, agree that humans burning carbon is the primary reason.

Vos Savant might respond, disingenuously, that because of Earth’s ever-changing climate, it’s implausible that humans are the only causal factor in any observed temperature change. Of course, this is correct. Put more simply, the planet’s temperature most likely would be different today than it was in 1880 even if we were still living before the discovery of fossil fuels. The problem with this argument is that humans might well be responsible for more than 100% of the observed change. In other words, without the heavy hand of industry, temperatures might just as likely be colder than they were 135 years ago rather than 1.4 degrees warmer.

Vos Savant’s dissimulation continues in her “solution”. Rather than focus on “attempting to maintain the [already severely compromised] world the way it is today”, which she claims is impossible given the climactic variations the earth has experienced over its 4 to 5 billion year history, she urges us to “develop more coping mechanisms for whatever nature alone may be handing us”. Vos Savant is silent as to the ways non-human animals and plants, upon which humans rely for our own survival, can adapt to climates for which they are wholly unsuited. Likewise, she appears dismissive of the extraordinary costs of protecting the billions of people living at or near sea level as the oceans continue their inexorable rise.

Due to her ubiquity in Sunday newspaper supplements and her *gravitas* as one of the world’s smartest people, vos Savant is undoubtedly influential. It is somewhat unfortunate that every now and then she makes careless math errors. It is highly regrettable that she ignores human responsibility for global warming, downplays the existential threat it poses, and promotes highly unrealistic approaches to living on our ever-hotter planet.

Wow Hal, you went a long way for that one.

Allow me to correct you:

What Savant’s problem states:

“(A) an annual $1000 raise or (B) a twice-yearly $300 raise?”

What you say:

“The problem is that vos Savant wrongly compresses the distribution of each $300 annual raise such that it’s paid in full in half a year”

You are wrong Hal. She doesn’t say a $300/year raise and then switches it to a $600/year raise with half granted after 6 months.

If you can’t understand a simple arithmetic problem why should anyone think you can understand or even discuss a complex scientific topic?

Jeff – if your boss tells you that you will be receiving a $5,000 raise starting January 1. How much of that raise will you have received precisely six months later?

N.B. – I am replying without any insults, snarky comments, or implications regarding your intelligence.

The problem is that vos Savant wrongly compresses the distribution of each $300 annual raise such that it’s paid in full in half a year”

Who said it was a 300 dollar annual raise ?

That’s where you’ve gone wrong it’s a twice a year 300dollar raise meaning so you have no reason to 1/2 the three hundred

Thanks for commenting Abdalla. Think of it this way. When you get a raise, it can be a raise in your hourly wage, it can be a raise in your weekly pay, it can be a raise in your monthly pay, or it can be a raise in your annual salary. Thus, you might get a $1/hour raise. You might get an extra $100/week. Maybe your monthly pay goes up by $1,000. If you’re told that you’re getting a $300 raise above your current $15,000/year salary that would have to mean that your new salary is $15,300 not $7,150 per half year. Right?

Are you sure she didn’t say $600 twice yearly vs $1,000 a year? That would make a better riddle since a little thought is required to realize that a twice-yearly $600 raise to your yearly salary means only an additional $900 paid out in a year. Just a thought.

$2500. If by boss tells me that I will receive a $300 increase every 6 months I would expect to receive an additional $50 every 6th month

Your math skills need some work.

With a salary of $30k year with $300 raise every 6 months shown in 6 month intervals

Year 1 =$15000+15300=$30,300

Year 2=$15600+15900=$3,1500

Year 3=$16200+$16500=32,700

Year 4 =$16800+$17100=$34,900

With a salary of $30k year with $1000 raise every 12 months shown in 6 month intervals

Year 1=$15,000+$15,000=$30K

Year 2=$15,500+$15,500=$31K

Year 3=$16,00+$16,00=$32K

Year 4=$16,500+$16,500=$33K

Now one could derive the equation and simply plug in the numbers and it would demonstrate the same thing. If you can’t understand the explicit example I doubt if you would understand the derivation.

Now you could have taken the time to figure out your error…I pointed it out. But you didn’t.

Now you can also falsely interpret her $300 raise every 6 months as a yearly salary increase of $300 every 6 months but that’s not what she said. When one gets raise of $300 one would expect to get an increase of $50 on next months paycheck ($50*6=$300) and again in 6 more months.

Even in the link you posted the author stated that Ms Savant was correct.

” Not that you’re going to catch me saying Marilyn vos Savant is wrong. On the contrary, her response is 100 percent correct. It’s just not necessarily the answer to the question she was asked.”

Smart people are very literal Hal. They don’t read things into statements that aren’t there. You (and the linked article) make the assumption that she meant:

“Given the question as stated, many people would interpret a $300 raise to mean a $300 increase in annual salary — that is, after six months your salary would rise from $10,000 to $10,300 per year.”

Many people would be wrong because that is not what she said. A $300 raise every 6 months is not the same as an increase in your yearly salary of $300 every 6 months.

So you use a misinterpretation of the question that allows you denigrate her intelligence and use that as the reason to proclaim her stance on global warming to be false. Should you be held to the same standard Hal? Should your opinion on one subject be dismissed because one could falsely interpret one of your statements on a different subject that suits their needs?

The twice-yearly $300 raise is a raise in your rate of pay, not a set bonus. So from January to June, your rate is $30,000 a year (or $15,000 for 1/2 a year), then from July to December it’s $30,300 a year (or $15,150 for 1/2 a year).

Jan to Jun: rate is $30,000, period is 6 months, so the amount paid is $15,000

Jul to Dec: rate is now $30,300, period is 6 months again, so the amount paid is $15,150 (1/2 of $30,300)

Therefore you make $30,150 that first year, not $30,300.

Vos Savant gives you the option of starting with a $30,000 annual salary and (1) a $1,000 per annum raise or (2) a $300 raise every six months. The raise obviously means a raise in the salary which is defined by a 12-month term. Under the second option, you get a $300 raise every six months to your annual salary. There is no other plausible explanation for a $300 raise to an annual salary but that the payout of that raise is over a year. Accordingly, each $300 raise equals $12.50 more per month. Since the raises occur twice yearly, if you select option (2) your wages effectively go up each year by $50 per month. In contrast, a $1,000 raise equals $83.33 more each month.

N.B. Jeff – In explaining the answer to you, I neither insulted you nor made any snide insinuations.

Your statement:

“The raise obviously means a raise in the salary which is defined by a 12-month term.”

is false. That’s how YOU interpreted it. Its not how Ms. Savant intended it because her solution is correct for her statement. YOU chose to interpret it in a way not stated.

I will refer you to the original question as you linked it:

“Suppose you make $10,000 a year. Your boss offers you a choice: You can have a $1,000 raise (not a bonus) at the end of each year, or you can have a $300 raise at the end of each six months. Which do you choose?”

It doesn’t say a $300 increase in your yearly salary every 6 months. It says a $300 raise every 6 months.

Even in your analysis you restate her question to suit your needs:

” Under the second option, you get a $300 raise every six months to your annual salary. ”

Go ahead and twist the language all you want Hal. The semantic squabble has nothing to do with her intelligence or her stance on global warming…no matter how much you want it to be so.

Okay Jeff. Can you explain how one could possibly interpret a $300 raise of a $10,000 annual salary as equivalent to a $600 annual raise?

Easily Hal.

Its right there in the question:

“Suppose you have a job offer with a choice of two annual salaries. One is $30,000 with a $1,000 raise every year. The other is $30,000 with a $300 raise every six months.”

As I said before…you can interpret the question any way you wish and make a big deal about it. I thought you didn’t like “gotcha” games? I guess it’s OK when the ends justifies the means. Her wording of a question has nothing to do with her stance on global warming. You really don’t want to go down that path do you Hal? In your world it seems that being wrong on one subject invalidates your stance on every other subject.

We’ll have (or not) to agree to disagree.

Jeff – you may disagree with me but Marilyn has acknowledged error. From her discussion board:

———

Recently on Marilyn’s discussion boards…

From imatechfan:

“I have been an HR professional for over 35 years. When a $30,000 a year employee receives a $300 raise, the annualized salary becomes $30,300. There is no other way of interpreting these words. In the first year, the employee would earn $30,150 and it would increase by $600 in each ensuing year. This makes the $1,000 annual raise is better.

If common terminology is used then the common definition/usage must be applied.

Also to The Dude: No big deal but “bi-annual” means every 2 years – you meant “semi-annual” which is every 6 months.”

Said perfectly!The error was a semantic one…not a mathematical one.’

She meant the question to read one way and you read the question another.

It doesn’t make her “wrong” and therefor your conclusion that she is wrong on global warming because her math was wrong is itself, wrong on that reason…not to mention all the others.

Jeff – In light of the fact that Marilyn herself admits she was wrong, do you persist in arguing that she might be right?

Admits she was wrong? What she admits to is that the question, as worded, could be read in two ways.

“If common terminology is used then the common definition/usage must be applied.”

Read it one way and the math is different that reading it another way.

Read it the way she meant it the math she did was correct.

Read it the way you did then the math you did was correct.

Why is that so hard for you to understand? Is it because to you, like Tom Brady, winning is everything?

Jeff – In arguing in favor of his solution (and mine) and that Vos Savant’s is wrong, imatech fan wrote: “

There is no other way of interpreting these words.“. In response, Vos Savant wrote: “Said perfectly!” (! retained). This is not an admission that the problem can be read in two ways. This is an admission that “there is no other way of interpreting” the problem except in a way that proves my (i.e., Vos Savant’s) “solution” in error.Trouble is Hal. She didn’t say “Said Perfectly!”

imatechfan did.

Once again your defense falls flat on its face.

http://marilynvossavant.com/forum/viewtopic.php?f=1&t=2036&start=15

Now what will you twist so you can declare victory?

You’re correct here Jeff. I misread the post. Marilyn has not admitted she was wrong. More evidence, actually, that she is not a credible source.

To you Hal, anyone that thinks for themselves is not a credible source. That much is painfully obvious to even the most casual observer.

I’ll make a deal with anyone willing: If you will send me $1000 at the end of every year, I will send happily send you $300 at the end of every 6 months. (The $30,000 salary is a distraction and does not need to be factored in. Start with zero. (If you send me $1000 once a year I will gladly send you $600 in exchange – and splitting into two payments is not a problem. What do you think? Deal?

Hal, right on the money! I thought the exact same thing when I read her column two weeks ago. It actually surprised me that she could’ve made such a basic mistake. But apparently people get befuddled when talking salary & raises, etc. It’s obvious from commenters like JEFF, that he can’t wrap his head around the discrepancy…. keeps crying about what she “meant” to say and its all about interpretation. Not!

JEFF — ever job I’ve ever had where a salary is involved is an ANNUAL salary. A raise increases your annual salary (or your hourly rate, but that also equates to an annual amount). So… the problem is Marilyn was calculating a semi-annual raise of $600 in her math… which is a $1200 annual raise.

Just LOOK at what you’d start with each year if you had two $300 raises a year:

Year 1: Annual $30,000 Semi-annual: $30,000

Year 2: Annual $31,000 Semi-annual: $30,600

Year 3: Annual $32,000 Semi-annual: $31,200

Year 4: Annual $33,000 Semi-annual: $31,800

etc.

It’s an obvious mistake… whether she admits it or not, whether you think so or not. I’m surprised that anybody could argue counter when faced with clear facts.

Let’s cut to the chase. It doesn’t matter what anyone “thinks” about why the planet is warming. Ask the scientists who consider the data without bias. And then, more importantly, connect the dots. Humans won’t stop dumping carbon into the atmosphere. Look at any balance sheet from any extractor, refiner, and distributor. We can’t get off of coal, oil, and gas for the same reason why we can’t get rid of Aetna. Corporations aren’t people. Corporations own people. The only way out of the contract is for most of us to die. And that includes their shareholders.

Thanks Terris.

Hey there! I’m at work browsing your blog from my new iphone!

Just wanted to say I love reading through your blog and look forward to all your posts!

Keep up the fantastic work!

I have read that question before, am an engineer with multiple degrees and can assure you that both mr. “jeff linder” and Marilyn are correct. I have been employed for over 38 years by various employers. No manager I’ve ever known would tell an employee that they will receive a $300 raise every six months and consider that to be a cute, HR correct, or perhaps “official” way of wording that your $300 annual raise will be distributed at a 6 month interval of $150. In fact, many raises are “Presented, worded, etc,” in their paycheck or hourly amount. ($100 a month, $1 an hour, etc). Do you think that means they take a $100 as annual and divide it by 12? No. No one would think that. The HR person quoted (only one in the world that I have ever heard of with that type of math) decided to also use the alternate, rare, interpretation that is not direct, not obvious, and not common. That is going out on a limb and requires a stretch, and a distorted interpretation, not a direct one. In most industries and businesses we speak to our employees using common English and math, and are not deceptive or misleading. There may be exceptions. Marilyn also is extremely polite and will honor and respect any alternate interpretations or views in her replies, and you will very rarely see her calling a reader outright wrong if they interpreted the question differently but she is more likely to jump at the opportunity to take the blame for it not being worded as well as it could have. In this case, I only know 2 or 3 people on the globe who could have misinterpreted that question and they are right in this thread.

I’ve been contemplating this problem for some time and think I’ve solved it.

The problem states that your income starts at $15000 which is your annual salary. For a 40 hour week for 52 weeks that breaks down to $14.423076 per hour. Let’s simply & say $14.42 per hour. (Keep in mind our figures will be approx. unless we use all the decimals)

If you get a #300 raise whether it’s monthly, bi-annually, yearly or even every 3 months, the increase is to be figured into your hourly rate by $300 divided by 2080 hours which comes out to $0.144232307. Let’s round it of to $0.14 additional per hour which makes it $14.56 per hour after the first raise.

Now for the first year you make $14.42 for 6 months & $14.56 for the other 6 months. That’s $14.42 times 1040 + $14.56 times 1040 = $14996.80 + $15142.40 = $30139.20 The other way is just $30000. The first method wins the 1st year.

Now for the 2nd year the increase at the beginning of the year and mid year is a plus $0.14 so the formula is $14.70 times 1040 + $14.84 times 1040 = $15288 + $15433.60 = $30721.60. The other way is $1000 divided by 2080 = approx $.48. So $14.42 + $0.48 = $14.90 times 2080 = $30992 which is over $270 more than the 1st way.

The $300 is a raise to be figured into your annual salary. If you figure it out the way Marilyn does that’s like getting a $600 increase in your pay twice per year which is why that way shows an increase annually of $1200. That’s not what a $300 raise is. Doing the above hourly method proves it.

Wayne

Correction on the previous comment.

I stated the initial income as $15000 should have been $30000.

Wayne