Order of operations: does it really matter?

George Orwell

While surfing the webpages of a variety of newspapers this morning, I stumbled on the following….

On the staged-reality-tv show Big Brother (UK version), they gave the housemates a mathematical task: they had to compute three different sums, then use the three resulting answers as the combination to a safe.

Implicit in the problem was that the three calculations should each result in a two-digit integer.

Video posted to the Channel 4 website shows the contestants muttering, struggling, and having an extremely difficult time of it. And with good reason!

Here are the three calculations they were given, as posted on the Channel 4 website:

Sum#1: 3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

Sum#2: 1396 x 2 ÷ 4 — (12²) + 46 x 2 ÷ 40 x (5²) – (7 x 99) x 3 – (11²) x 5 – 219

Sum#3: 100 – 33 x 5 + 665 ÷ (5²) x 17 – 248 x 3 ÷ (4²) + 52 ÷ 7 + (273 – 217)

From the look of things on the on-line video, I’m guessing that the contestants had no writing implements, and had to do all of this in their head. That makes this challenging enough, I suppose.

Making matters worse is that none of the three sums is integer valued; they work out to 62/5, -4583/2, and 26189/70. Rather, these are their values if one computes using the usual order of operations, where exponents have precedence over other operations, where multiplication and division take precedence over addition and subtraction, where calculations are performed left-to-right, and parentheses can be used to override this sequencing. (“PEMDAS” is a popular acronym with my students, standing for “Parentheses, Exponents, Multiplication and Division, Addition and Subtraction”, and sometimes recalled using the mnemonic “Please Excuse My Dear Aunt Sally”)

Apparently the folk who created this puzzle expected their contestants to work left-to-right, ignoring operator precedence, in the way that a $1 calculator might do. (Calculators that do pay heed to order of operation conventions are often marketed as “scientific” calculators.)

For example, the first sum should go as follows:

3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

= 3 x 17 – 24 + 78 x 9 ÷ 5 – 169 + 36÷ 4 + 16 – 21 + 9 + 99 – 49 – 49

= 51 – 24 + 702/5 – 169 + 9 + 16 – 21 + 9 + 99 – 49 – 49

= 62/5

But I suspect the intended calculation was instead:

3×17 = 51, 51-24 = 27, 27+78 = 105, 105×9 = 945

945÷5=189, 189-(13²)=20, 20+(65-29)=56. 56÷4=14,

14+(4²)=30, 30-(7 x 3)=9, 9+(3²) + 99 – (7²) – 49 = 19

Similar (incorrect!) computations for sum #2 and sum #3 yield 31 and 75, respectively.

Clearly it is important that we agree on our order of operations.  But why do we prefer one over the other? Is this merely a cultural convention?  One stock answer to this is to point to the algebra of polynomials: our conventions regarding operator precedence play a central role in how we interpret linear equations (e.g. what is the slope of the line ?), how we interpret polynomials (e.g. is a quadratic or a cubic polynomial?), and how we compute sums and products of polynomials.

But this morning, having not yet had my first cup of coffee, I wonder: is it possible to change the rules of arithmetic, so that all operations have the same precedence (unless exceptions are forced by parentheses), and to develop a meaningful algebra based on similar principles? It seems to me that the answer is yes, and I wonder exactly what is lost by doing so, other than familiarity.

The Trouble with Units

There have already been a couple examples of what can go wrong when you mix up Metric and Imperial units on a Boeing 767 or a spacecraft to Mars. But it turns out that even nonstandard units can cause a little bit of trouble. By “nonstandard units”, I mean units such as spork and by “little bit of trouble” I mean an accounting error of $66,500,000. That’s a lot of cutlery.

The error is related to the aftermath of Hurricanes Katrina and Rita. Back in 2005, a large number of house supplies (pots, pans, toilet paper, etc.) were donated or purchased for people who had lost their homes. The items were stored in a warehouse in Louisiana, and then in Texas, and then…nothing. They sat there in Forth Worth, and FEMA paid about a million dollars a year to keep them stored. And not long ago FEMA decided to give them away rather than continue to pay that storage (plus the warehouse was apparently going to be torn down). At this point the story became quite public, and Louisiana said that HEY they still needed those, because three years later there are still people who are recovering, and why have all these things been collecting dust anyway? And FEMA said, Well we offered them to you and you didn’t want them, so don’t blame us. Etc.  Etc. And during all this arguing, the reported value of the supplies was listed at $85 million.

But now it turns out that it wasn’t $85 million after all — it was more like $18.5 million. Because when they were counting things, they counted single items such as  one spork the same as entire cases, as in one case of sporks. As the General Services Administration explained last week, “The final adjustments reveal there was a significant overstatement in the total asset valuation.”

So on the one hand, this is good because that’s $66.5 million dollars that wasn’t wasted after all (indeed, it never existed). And on the other hand, the fact that there was such a large accounting error on top of the revelation that all this stuff was just sitting there doing nothing isn’t really making anyone feel very good.

(See CNN for more details.)

We’re guest posting over at Puntabulous!

Head over to Puntabulous for Teach Me Something Tuesday #14: A History of Cruises (since Craig is gone for the week on a Cruise Ship). One of the facts that arose in the comments this morning was about the gas mileage of Cruise Ships. If you’re worried about your 20 miles per gallon and envying a 40 mpg hybrid (even though it might not be as good a deal as it sounds initially), be glad that you’re not taking the Queen Elizabeth 2 to work. The QE2 gets only 40-50 feet per gallon.

The Queen Mary is even worse, getting 13 feet per gallon*. That’s .0025 miles per gallon or, put another way, over 400 gallons per mile. And if you’re talking gallons per mile instead of miles per gallon, you know these rising fuel prices are hitting you especially hard.

*13 feet per gallon = 1 meter per liter, which is fun to say out loud.

Mathematician-of-the-Week: Pierre-Joseph-Étienne Finck

Running time of the Euclidean Algorithm

Pierre-Joseph-Étienne Finck was born October 15, 1797, and died July 27, 1870. Finck’s most significant mathematical contribution appears to have been his analysis of the running time of the Euclidean Algorithm, which he published in 1841.

One wonders if his own life experiences contributed to his interest in recursive algorithms. Upon graduating from the École Polytechnique in 1817, he was admitted to the Artillery School. However, he wasn’t satisfied with his studies there, and applied ( in March 1818 ) to transfer to the Royal Guard cavalry. Request denied.

4 months later, he applied to the cavalry again, this time saying that he would resign if his request was not honored. Request denied.

So he resigned from Artillery School…. But by early 1819 he had second thoughts, and applied for reinstatement to the Artillery School. Request denied.

At this point, he changed tactics, and began studying mathematics at the University of Strasbourg. He completed his doctoral dissertation (on movements of the terrestrial equator) in 1829. Ironically, by that time, he had been appointed as a mathematics instructor at the Artillery School of Strasbourg.

I suppose Finck’s life might provide a valuable lesson in the importance of sticking to your guns.

Source:  Mactutor History of Math archive

Carnival of Mathematics #37 at Logic Nest

The 37^th Carnival of Mathematics is up over at Logic Nest. It features prime-generating functions and integrals all the way to cats and Fourier series, with lots of interesting posts in between!

The host, Logic Nest, is the personal blog of Systems Analyst and family guy Ian Luke Kane.   Carnival aside, there are several posts on his front page that caught my interest, like the Pirahã in Brazil who have no concept of precise numbers, and a link to a story about why tape tears.

Forty three, or McNugget Numbers

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I am 43 years old, as of today.  

I don’t normally think much about such things (turning 40 was an exception to that one), so I’ve been surprised by the way “43″ has impressed me, looming on the horizon.  I was trying to figure out what was going on with that, and eventually realized it was the fact that 43 seemed like such a dull number.

So I set out to discover cool facts about the number 43.

There were a few obvious things:  43 is prime, for example [but there are LOTS of those].  It can be written as a sum of consecutive integers in two ways (21+22, or -42+-41+…+42+43), but again, no big deal there.

Inspired by a recent observation that no matter what base your number system is, you’ll always describe it as being base 10 [akin to the old joke: "There are 10 kinds of people, those who understand binary and those who don't."], I was playing with alternate bases — maybe 43 has cooler properties in some other base.  The best thing I’ve found along that path is .  How cool is that?   Well, maybe not the coolest thing ever, but I now have a new favorite piecewise linear function: (n base n).

And then I finally found some cool math involving 43!  The Wikipedia page on 43 pointed out that 43 is the largest natural number that is not a McNugget Number; that is, you cannot buy 43 McNuggets at McDonald’s by buying the usual packages of 6, 9, and 20 McNuggets each.  But you can get 44, 45, 46, … and all larger numbers of McNuggets.  For example, 44 = 20 + 24 = 20 + 6×4, so you can order four packages of 6 McNuggets, and 1 pack of 20, and have your 44.  45 is five nine-packs, 46 is a sixer and two 20s.  But if you need to purchase exactly 43 McNuggets, you’re out of luck.

One way to verify that 43 cannot be so gotten:  It is easy to tell if a small number is a McNugget number (e.g. it is obvious that 8 is not).  And note that if a larger number is a McNugget number, then subtracting 6, 9, or 20 should give another McNugget number.  So we start by letting A = {43}, and at each stage we replace each member x of A with x-6, x-9, and x-20.  If at any point we get a member of A that is a McNugget number, reversing that path would show that 43 is also a McNugget number. (And conversely, if 43 were a McNugget number, such a path would exist.)

We have: 

Initial step:  {43}

Second step: {37, 34, 23}  [so e.g. if 34 could be built out of 6, 9, and 20, then adding 9 gives 43]

Third step: {31, 28, 17, 28, 25, 14, 17, 14, 3} = {31, 28, 25, 17, 14, 3}  [we can ignore 3, obviously not a McNugget number]

Fourth step: {25, 22, 11, 22, 19, 8, 19, 16, 5, 11, 8, 8, 5} = {25, 22, 19, 16, 11, 8, 5} 

Fifth step: {19, 16, 5, 16, 13, 2, 13, 10, 10, 7, 5, 2, 2} = {19, 16, 13, 10, 7, 5, 2}

Sixth step: {13, 10, 10, 7, 7, 4, 4, 1, 1} = {13, 10, 7, 4, 1}

Seventh step: {7, 4, 4, 1, 1} = {7, 4, 1}

At this point we can probably stop, as we clearly cannot build any of these three numbers as sums of sixes, nines and twenties, and hence we cannot build 43 as such a sum either.

Proving that every number greater than 43 is a McNugget number is a nice application of generalized induction:  one shows that 44, 45, 46, 47, 48, and 49 are McNugget numbers as the base case (by direct calculation for each).  The inductive step is to assume that n > 49 and that every number between 44 and n-1 is a McNugget number.  Then n-1 > n-6 > 43, so n-6 is a McNugget number, and hence n = (n-6) + 6 is another McNugget number.  Thus every number greater than 43 is a McNugget number.

Okay, I’ve decided 43 is cool after all. 

Perhaps I’ll celebrate with a birthday lunch…

4, 6, 8, 10, 12, 14,….What comes next?

The answer, naturally, is 15. If you’re talking about the Burnt Pancake problem, that is. (And the sequence actually starts 1, 4, 6, 8… but I left off the initial 1 because otherwise you would have known right away that something was amiss.)

The Burnt Pancake problem involves pancakes of different sizes, each with one burnt side, piled up on top of one another. Here’s how famous math guy and Emmy winner David X. Cohen initially described the problem in this interview with Sarah Greenwald:

The question was how do you sort these disks to get the biggest pancake on the bottom and the smallest pancake on top [with all the burnt sides down] if they start in an arbitrary disordered state, and the only thing you’re allowed to do is put a spatula somewhere in the middle, pick up the ones above it, flip them over, and put them down, as a group. Doing that repeatedly, putting a spatula in different places, you want to sort this out. So a very physical thing, that got me excited when I found out that no one knew the answer in general for how many flips it takes to sort this thing.

Here is help us visualize this problem is our friendly neighborhood Godzilla. He’s going to use oreos with tops removed to simulate the pancakes.

With only one “pancake”, if it starts like this

the “burnt” side is already at the bottom, so it needs 0 flips to get into the proper position.

If, instead, the pancake starts “burnt” side up, it needs one flip:

(Don’t you think that Godzilla looks a little bit like that guy Craig from Hell’s Kitchen?) As the Big G has just demonstrated, if you have just ONE burnt pancake then it could take as many as, well, 1 flip to orient it correctly. That’s where the 1 in the sequence 1, 4, 6, 8, … comes from.

Now let’s look at what happens with two burnt pancakes. We want to end up with the larger pancake on the bottom, and all the burnt sides down like this:

But what will happen if the pancakes start off in a different configuration? How many flips have to be done? It turns out that it could be as many as 4 flips:

Suppose your pancakes start off in this pile: , with the burnt parts on top. The first thing Godzilla does is to put the spatula on the bottom and flip the pile over.

Now the pancakes have the burnt side down, but the smaller one is on the bottom. The big top pancake will have to be flipped:

(Godzilla’s being a little sloppy here: he should only be picking up the top pancake.) After this maneuver the burnt parts are on the “outside” but big pancake is still on top. The entire stack needs to be flipped to get the little pancake back on top:

Now the little pancake is on the top and the burnt parts are on the “outside”, so the top pancake must be flipped:

And now voilà, the pancakes are in the right order!

See how happy Godzilla is! He knows he gets to eat these oreos when all is said and done.

With a different starting configuration (there are 2!·2²=8 ways they initially could be piled up, with the smaller pancake on the top or the bottom, and the various burnt sides up or down), it turns out that it will take at most 4 flips to get them in the correct order. That’s where the 4 comes from in 1, 4, 6, 8, ….

What happens if you start with three pancakes of different sizes? The desired ending configuration is this:

There are 3!·2³=48 different ways the pancakes could start out. Some of them could be turned into the proper configuration after just one flip:

but other configurations require more. This one, for example:

The cookies are in the right order, but it just takes a lot of maneuvering to get the burnt parts on the bottom instead of the top: 6 flips. That’s the most it would take no matter how the pancakes started out.

So for one pancake it could take as many as 1 flip, for two pancakes it takes up to 4 flips, for three pancakes up to 6 flips, for four pancakes up to 8 flips, for five pancakes up to 10 flips, for six pancakes up to 12 flips, and for seven pancakes up to 14 flips.

Then for eight pancakes, it only takes up to 15 flips. And for nine pancakes 17 flips, then for ten pancakes it goes up to 18 flips (according to the Online Encylopedia of Integer Sequences). But by 11 pancakes there are over 81 billion different initial configurations, so checking by hand to find the smallest number of flips for each configuration is tough. Fortunately, as described in this earlier post, we have E. Coli to help us figure it out. [They give the E. Coli a configuration, let them do a specific number of flips, and any E. Coli that get the virtual pancakes in the right order become resistant to the antibiotic tetracycline in honor of their good effort. Then the scientists add some tetracycline and if any of the E. Coli survive, they know that the pancakes could be put in order after that particular number of flips. The animation E. Hop gives the whole scoop.]

Meanwhile, Godzilla is going to take a little break. Bon appetit!

Math Mistakes in the News: Calculator Time

I found this story on Eric Berlin via God Plays Dice.

The Herald reported last week that a Traffic Warden was incorrectly ticketing cars in a Devon, England parking lot because of how he was using a calculator. In this parking lot, drivers would pay for a certain amount of time and then post a slip in the windshield with the time they’d entered and how long they’d paid for. One driver, for example, entered at 2:49pm and paid for 75 minutes.

Now 75 minutes is 1 hour, 15 minutes so the driver was covered until 4:04pm. But the Traffic Warden figured out the expiration time by entering in 14.49 into his calculator (for 1449 military time, which corresponds to 2:49pm) and adding on 0.75 (for the 75 minutes). He got 15.24, which he interpreted as meaning that the driver was only covered until 3:24pm. Since it was already 3:41pm, he issued the car a ticket. The car owner saw all this and tried to explain the error — that hours have 60 minutes, not 100, so standard decimal addition doesn’t apply — but the Traffic Warden didn’t see any problem and continued to ticket cars.

In good news, after appeal the incorrect tickets were repealed and a letter of apology sent.

Happy Pi Day!

Happy Pi Day!!!

22 July is celebrated throughout (much of) the world as Pi Day, for the ratio 22/7 is a reasonably accurate rational approximation to the number π.

Pi Day is also celebrated on March 14, in those parts of the world who would abbreviate today’s date ( July 22, 2008 ) as 7/22/2008, since March 14 becomes 3/14 under such a scheme. According to Wikipedia (“So you know it’s true!”™), only a handful of countries follow this scheme. Most would abbreviate using either a little-endian scheme ( 22/7/2008 ) or a big-endian scheme ( 2008/7/22 ). The amount of space on Wikipedia devoted to a flamewar discussion about the relative merits of each scheme is astounding.

There are many days when I’m happy to be a mathematician, and not a copy editor for an international open content network based encyclopedia.

Fun with Bases

In The Hitchhiker’s Guide to the Galaxy by Douglas Adams, the computer Deep Thought gives the Answer to the Life, the Universe, and Everything as 42. Google calculator gives the same result.^* (Of course, that’s not much use without knowing what the question is.) At the end of the later book The Restaurant at the End of the Universe, a caveman pulls out scrabble tiles and forms the question, “What do you get if you multiply six by nine?”

Of course, the universe is an imperfect place, and six times nine is actually 54. At least in Base Ten. In Base Eleven, 6×9= 4A (where A stands for the digit TEN, since fifty-four is 4 elevens and A=ten ones left over). And in Base Twelve, 6×9=46, since fifty-four is 4 twelves and 6 ones left over. And finally, Lo and Behold, in Base Thirteen, 6×9=42 since fifty-four is 4 thirteens and 2 ones left over.

So maybe the universe isn’t such a wacky place after all! Maybe Douglas Adams was secretly suggesting that we should all be using Base Thirteen. Or maybe not: when Douglas Adams was asked about this, he supposedly replied, “I may be a sorry case, but I don’t write jokes in Base 13.”

^*Seriously. Try typing in “answer to life, the universe, and everything” all lowercase.

Thanks to Anya for pointing this out!

Mathematician of the week: Georg Pick

Georg Pick was born on August 10, 1859, and died on July 26, 1942, having spent much of his career as a professor at the German University of Prague.

He is most commonly remembered for a theorem concerning the area of lattice polygons, polygons in the plane whose vertices occur at points with integer coordinates: the area of such a polygon is numerically equal to one less than the number of lattice points in its interior plus half the number of lattice points on its boundary. Pick published his theorem in 1899, but it received scant attention until it was commented upon by Steinhaus in his 1969 text Mathematical Snapshots.

Part of the popularity of Pick’s Theorem is its elegance: it is simple to state, it is simple to discover (once told to expect a relationship involving area, lattice points in the interior, and lattice points on the boundary, initial explorations with rectangles suffice to generate linear relationships that suggest Pick’s result); and a formal proof [e.g., this one at Cut the Knot] is relatively straightforward.

Mathematicians with significant anniversaries during the week of July 20 – July 26:

July 20: Death anniversaries of John Playfair [1819] (Playfair’s Axiom), Bernhard Riemann [1866] (geometric foundations), and Andrei Andreyevich Markov [1922] (probability, stochastic processes)

July 21: Birthday of John Leech [1926] (Leech Lattice); death of Giovanni Frattini [1925] (group theory)

July 22: Birthday of Wilhelm Bessel [1784] (analysis), Gabriel Lamé [1795] (differential geometry, proof of FLT for exponent 7), and Konrad Knopp [1882] (analysis)

July 23: Death of Florence Nightingale David [1993] (statistics) [her parents were friends of "the" Florence Nightingale]

July 24: Birthday of Errett Bishop [1928] (Foundations of Constructive Analysis); death of Hans Hahn [1934] (Hahn-Banach Theorem)

July 25: Birthday of Johann Listing [1808] (topology)

July 26: Birthday of Kurt Mahler [1903] (p-adic numbers, geometry of numbers); deaths of Gottlob Frege [1925] (mathematical logic), Henri Lebesgue [1941] (measure and integral), Georg Pick [1942] (Pick’s Theorem), Raymond C. Archibald [1955] (history of mathematics), and John Tukey [2000] (mathematical statistics)

Source: MacTutor

It’s a bird! It’s a plane! Oh, wait, it IS a plane.

This news story made the rounds many months ago, but I didn’t read of it until I was paring down my Inbox this week (1388 messages. It was getting a little overwhelming) and found it in a news digest. In late March, the Japan Aerospace Exploration Agency accepted a proposal of a project led by Shinji Suzuki to make origami spacecrafts and launch them from the International Space Station. How cool is that?

One worry is that they would burn up because friction from entering the atmosphere tends to make things rather hot, but it’s possible/likely that they won’t both because of their shape and because they will be traveling so slowly through the atmosphere (plus the paper, made from sugar canes, is heat resistant). When I first read this I envisioned Giant Origami Planes, but they’re actually small: the shuttles will only be 8 inches by 4 inches after folding, and weigh just over an ounce.

Another worry is that there is no way of controlling where they land, or even how to track them. This is a much bigger deal, and perhaps one reason they’re not going with the Giant Origami I’d envisioned (can you imagine if one of those swooped down onto your lawn?). But don’t go looking too soon: the grant they received is for 3 years of feasibility studies.

While you’re waiting, you can learn how to make an Origami Rocket.

I read this story in many places, but got most of the info for here from Discovery News.

The Math of Language

My cousin Taimi is a linguist, and at our Big Family Reunion last month she told me that some people tried to develop a mathematical symbolism for language (language in general, regardless of the actual language spoken) and it worked well for speech that was meant to be informative. For speech patterns that were social and culturally based, though (when and how to thank someone, for example), the math language of language fell apart. It just couldn’t be applied universally correctly.

So when we returned to New York I looked up what this might be and found myself, well, confused. The closest example that I could find (which may or may not be what Taimi was referring to) was in a Keith Devlin column from 1996. He used an example from X-bar theory. (Doesn’t X-bar sound like a drinking establishment? In fact, there is such a place in Los Angeles, although it looks geared more towards Gen X than mathematicians).

In linguistics, X-bar theory seems to be a way of describing all sorts of phrases in a recursive fashion. So if you want to talk about the noun cat, you might add on some descriptions like gray or fabulous. The grammatical rules allow you to change X (a noun, a verb, a proposition, an adjective, etc.) into something that’s modified with things called complements and adjuncts, and the result is called X-bar and should be written but is often written just as X’ because it’s hard to typeset the whole “bar” thing. So a noun-bar (N’) would be fabulous gray cat (instead of just cat) or food bowl (instead of just bowl)

Then you can move up to X Phrases. An X Phrase is an optional specifier, followed by an X’, and then maybe some Y Phrases. This is written along the lines of
XP→(specifier)X’YP*

As an example a noun phrase (NP) would be something like “the fabulous gray cat” where the is the specifier and fabulous gray cat is N’. Another noun phrase is “the food bowl” where the is a specifier and food bowl is N’.

From this you can form a verb phrase (VP) “sees the food bowl”. Here the V’ is just the single verb sees but it’s followed by the noun phrase the food bowl. In other words, VP→(specifier)V’NP*. This is where the recursive part comes in, using Y phrases to build X phrases.

Presumably the next step would be to combine “the fabulous gray cat” and “sees the food bowl” into an actual sentence, but building sentences involves a whole other set of rules.

In many ways this reminds me of diagramming sentences, except that instead of starting with the sentence and breaking it down, the rules have to be developed in such a way that they can be described regardless of the actual sentence or even actual language.  Because the beauty and complication of this is that all of the X-bar rules apply regardless of whether your noun phrase is “the fabulous gray cat” or “el gato gris fabuloso” or “η μυθική γκρίζα γάτα”.

How big is an Acre? How much is inside other stuff?

How big is an acre? It is, of course, 4046.9 square meters, although that 0.9 is shorthand for 0.8726… if you’re surveying in the USA and shorthand for 0.8564… if you’re not. But how big is that? If the answer of “four roods” doesn’t help you, then a popular comparison is that it is about 3/4 of a football field:

Created by Xyzzy n and posted under GNU Free Documentation License

It was in searching for a visual description of an acre that I ran across the great site How Much is Inside? (with translations into Arabic, Dutch, German, Spanish, and Swedish). Rob Cockerham and his crack team of reporters tackle all sorts of questions about non-standard measurement. For example,

• How much gold is inside a bottle of Goldschläger?
• How many CDs can be labeled with a Sharpie® before it runs out of ink? (more than you think!)
• How much actual granola is in a box of granola bars?
• And, of course, How much is inside an acre?

As a bonus, there’s a Science Club which does fun things like drop toast to see if it usually lands butter side down and dissect Hot Pockets. Hooray for science!

Make a guess, any guess

There was an interesting article in The Economist a few weeks ago about how to get an accurate estimate of a not-necessarily-well-known quantity (e.g. How many people were on board the RMS Titanic on its premiere voyage?). One way, which was already examined by James Surowiecki is to ask a bunch of people rather than just one. In general, the too-high guesses and too-low guesses start to cancel out, and the average number is a pretty good estimate, or at least more accurate in general than only asking one person.

But what if you don’t have a crowd? Then Edward Vul and Harold Pashler said that you can just answer twice. Indeed, whether you make two guesses one right after the other or two guesses several weeks apart, the average will (on average) be more accurate than the original guess. Better accuracy comes when the guesses are several weeks apart, but several commenters note that this may be because people looked up the answer in the meantime. Edited to add that they, like I myself, should have looked more closely at the original article since (as the eagle-eyed readers below note) the authors point out that this is unlikely because the second guesses were typically worse than the first guesses. They just averaged to something that was a bit better.

Let’s simulate this. I asked four random people to make two guesses as to the number of people who sailed on the Titanic . For comparison, Wikipedia says there were 2223 people aboard that night.

Person #1 First Guess: 900
Person #1 Second Guess: 2300
(Average: 1600, closer than the first guess!)

Person #2 First Guess: 2000
Person #2 Second Guess: 5000
(Average: 3500, a lot worse than the first guess)

Person #3 First Guess: 3200
Person #3 First Guess: 1400
(Average: 2300, nearly exact!)

Person #4 First Guess: 4000
Person #4 First Guess: 5000
(Average: 4500, worse than the first guess)

The average of the first guesses was 2525, which was more accurate than every first guess except the guess of 2000. This illustrates the idea that the average over crowds tends to be more accurate than an individual.

Now for the purposes of illustrating this article, the average guesses should be a little more accurate than the initial guesses. This happened for two people, but not for the other two: the average Average guess was 2975, which is less accurate than 2525. Dang, real data is so uncooperative. Of course, Vul and Pashler asked 438 people instead of 4, so maybe if I had asked more people I’d have gotten a result similar to V&P.

[You can see what seems to be the original article from Psychological Sciences here.]