Leap year! What is it good for? Absolutely nothing? Or does it actually serve a purpose?

Okay, let me explain how this question came about... 

Today our little girl turned one year old. She was born at 1:09 in the afternoon on February 11^th, 2015. I mentioned this to my Algebra students today, and then added that at about 7:09 this evening, she will have completed one entire revolution around the sun.

They looked at me like I had a third eye growing out of the middle of my forehead (okay, they often look at me that way...). "If she was born at 1:09 in the afternoon, isn't that when she's finished her first revolution around the sun?"

Nope! You see, a year is not exactly 365 days long. A year (the amount of time it takes for the earth to travel once around the sun) is actually a little bit longer than 365 days. It's right around 365.25 days. So our little girl finished her first trip around the sun a quarter of a day after her birth hour. Since a quarter of a day is 6 hours (²⁴/₄ = 6), and 7:09 is six hours after 1:09, it was this evening that she finished her first trip around the sun.

This led the class into a discussion of leap year, because that quarter of a day is the reason why leap years exist. If our calendar year is a quarter of a day shorter than a solar year (a solar year is the time it takes earth to travel once around the sun), then every four years, we'll end up with an extra day that needs to be stuffed into the calendar somewhere. So we decided to make that day be February 29^th. It was actually very good thinking - since February was by far the shortest month, putting that extra day in the calendar in February helps to even out the months a little*.

"Of course," I added, "It's not exactly that simple, because a solar year is not exactly 365.25 calendar days. It's just a little bit less than that. Which is why, every hundred years, we skip a leap year."

"WHAT?"

"Oh yes. If the year number is divisible by 100, we don't have a leap year."

"Really?"

"Sure. Unless, of course, the year is also divisible by 400. In that case, we do have leap year."

It was at this point that one of my students suggested that Leap Year might be good for absolutely nothing. Actually, what she said was, "I don't see why we need to do all this math. Why can't we just forget leap year?"

Ah. Good question. Let me try to explain it to you. Take a look at the picture below. In it you'll see the sun, and planet earth. Earth is following that almost-circular orbit around the sun, and I've marked the position of the earth with the time: the time of my little girl's birth.

When the earth is in this position, we are essentially in the dead of winter.

Now, where will the earth be 365 days after this point in time? I'll show you. In the picture below I've marked 1:09 PM on Feb 11, 2016. That's the second line I've drawn. "That's it?" you might think. "Why that's hardly any difference at all!"

Sure, that's true. But that tiny little difference (about a quarter of a degree every year) will add up as the years pass.

Suppose we stop having leap years. And suppose that my little girl lives to be 120 years old (hey, who knows what scientific advances will happen in her lifetime?). When she's 120 years old, all those quarter days will have added up to an entire month. She'll be celebrating Christmas on December 25th, but the weather will be like Thanksgiving weather instead of Christmas weather, because the calendar year will have shifted an entire month from the solar year - and it's the solar year that determines our seasons, regardless of what we declare them to be on our calendar!

Can you imagine? She'll say to her great-grand kids, "I remember when we used to have blizzards on Christmas..." and they'll say, "That's silly, gram! We never have snow like that in December!"

And then, if we keep on like this, all those quarter-days will add up to 2 months, then 3. Eventually, our calendar year will be completely backwards from the solar year. Students will have June, July, and August off from school, and it will be snowing and bitter cold all through "summer" vacation. Then they'll be going to school all through the wonderfully warm and sunny beach weather of December, January, and February.

"Okay," my student said, "We can keep leap year."

I thought you might feel that way.

* Professor Puzzler's father, commonly known around here as "Mr. Physics," adds the following leap-year trivia:

"There is another reason whey they choose February for the extra day. Remember, under the Julian calendar the year started at March; February was the last month of the year. That is why September (7), October (8), November (9) and December (10) are numbered wrong.

"Another interesting bit of trivia. When Caesar Augustus decided that he needed a month named for him (August) he realized that to show himself equal to his Great Uncle Julius he needed to have 31 days in August so he stole a day from February leaving it with only 28 days."

I heard someone talking about the "Diameter of the Internet" recently. I understand finding the diameter of a circle; it's just the distance across its widest point. But what is the diameter of the internet, and how do you measure it?

Note: this item was originally posted on the "Ask Doug" blog* back in 2005. We've updated it and republished it with additional commentary at the bottom.

Have you ever played the game "Six Degrees Of Separation"? You start out with a person (often a movie star), and then see if you can relate them to another movie star in six or less connections. For example, how would you connect Julia Roberts to Sean Connery? Well, Julia Roberts starred with Richard Gere in "Pretty Woman", and Richard Gere starred with Sean Connery in "First Knight". That's two degrees right there.

The internet diameter is similar. How many degrees of separation are there between any two randomly selected web pages? How many clicks does it take to get from one web page to another? Researchers have done studies and calculations to try to determine the answer to this question, and the average "distance" in clicks between any two web pages is known as the diameter of the web. As of the turn of the millenium, the diameter of the internet was approximately 19 clicks.

And here we are in 2016, and you'd think we'd have updated statistics for you.

We don't.

We have some wrong statistics, though. Because, like everything else on the internet, people read what they want to read, and interpret things how they want to interpret them, with no regard for reality.

For example, in 2013, Smithsonian Magazine's website (yes, you read that right - I'm accusing Smithsonian of publishing bogus statistics - although, to be fair, they did partially retract their article) stated this information as though it was new, and reworded it to say that the maximum distance in clicks was 19. (Go back and read the original post; 19 was the average distance, not the maximum distance).

Others have made even more remarkable claims, such as "The diameter of the internet will never exceed 19."

But the reality is, nobody seems interested in continuing this line of study, and there doesn't appear to be any current statistical analysis on the diameter of the internet. So I guess we might as well keep saying the diameter is 19, even though that statistic is now about 16 years old!

There are some questions that would be fascinating to explore (if only I had more time!). Questions like:

• How do you deal with a website that is completely isolated (no inbound or outbound links)? Surely their distance to other websites isn't defined to be infinite? Because that would be problematic for averaging purposes! I suspect sites like that are dismissed from the equation.
• Did you realize that "distance" is not a "commutative" property? In the real world, if we say the distance from A to B is five miles, that means the distance from B to A is also five miles. But in the web, between any two points there are two distances - the number of clicks by outbound links from A to B, and the number of clicks by outbound links from B to A. These two numbers can be significantly different. To see why this is so, consider that there are thousands of websites around the world that link to The Problem Site. This gives a distance of one click. But The Problem Site doesn't link to most of those sites, which means the distance has to be more than one click. In fact, we should really think of it like distances between two cities when all the roads are one-way streets, so you have to take a different route from A to B than you would from B to A. So how do you calculate the distance? Do you take the smaller number? The larger one? Or the average?
• How are search engines included in the mix? Technically, Google links to virtually every website. But the pages that link to various sites are not static, because they are dependent on user input to display the links. On the other hand, if I search for "The Problem Site" on Google, and post the link here, I've now provided an indexable link for people to use when calculating the internet diameter. So have I changed my click distance to google by doing that?

Those are the questions that keep me up at night when I'm pondering 16-year-old statistics that no one else seems to care about any more! 

Professor Puzzler

* The "Ask Doug" blog was the precursor to the Ask Professor Puzzler blog. We are gradually moving the content from that blog to this location. Most items will be kept to their original publication dates, but occasionally, when we find an item of special interest, we'll re-post it with the current date, so our visitors will be more likely to find it!

A video has been floating around that talks about "fast thinking" and "slow thinking," and a simple three-question test that people who perform well on tend to do better on a lot of other tasks. Here are the three questions:

1. A baseball bat costs a dollar more than a baseball. Together they cost $1.10. What is the cost of the baseball?

2.  If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

3.  In a lake, there is a patch of lily pads.Every day the patch doubles in size. If it takes 48 days for the patch to completely cover the lake, how long would it take to cover half the lake?

The video doesn't give answers or explain all the questions. Can you help?

I'll give it a go. But first, let me guess what the "fast thinking" (incorrect) answers are:

1.  The ball costs ten cents.

2.  It takes 100 minutes.

3.  It takes 24 days.

Were those the answers you came up with? If they were, don't feel bad - these are the "natural" answers. But let's look at the real answers.

Baseball Problem

If b is the cost of the baseball, then the cost of the baseball bat is b + 1.  This means that b + (b + 1) = $1.10. This leads us to 2b + 1 = 1.1, or 2b = .1. Dividing both sides by 2 gives us b = $0.05, or five cents. This is half of the instinctive response.

Widget Problem

You have 20 times as many machines, which means they can make 20 times as many widgets in five minutes. You know what that means, right? It takes 5 minutes!  All 100 of those machines are working simultaneously, and each group of 5 machines makes 5 widgets in that five minutes.  This is similar to the cat and mouse problem which can be found in our reference pages: Cat and Mouse Variation.

Lily Pad Problem

The correct answer to this is not half of 48; it's one less than 48. If the lake is half covered on one day, it'll be completely covered the next day. Therefore, it takes 47 days for the lake to be half covered.

The following picture (click it for a larger version) was seen on Facebook by my outlaw Jakob (his wife is my sister-in-law, which is why I refer to him as my outlaw). It was accompanied by the caption "Here Is The Reason Why 1 Is Called 1. , And 2 Is Called 2..."

Somehow, Jakob must have known that I would want to write a sarcastic, cynical post about this.

So let's start with the caption.

"Here Is The Reason Why 1 Is Called 1. , And 2 Is Called 2..."

Uh...no...your terminology is confused. "1" is not what the number is called; it's called "one." "1" is the grapheme for the number we call "one."

So I guess you meant to say "Here is the reason we use this grapheme for one, and that grapheme for two..."

Better?

No. Not at all. Historically, there is absolutely no reason to think that people invented their numerical graphemes based on the number of angles in the digits (mind you, there is some indication that various civilizations played with creating graphemes based on making the number of lines match the numeral being represented). Most of our digits went through a lengthy evolution process, and their original forms, like our modern forms, had far more curves than angles. Take the number three, for example. The number three has no angles at all. Neither does six, eight, or nine.

"Well sure," you might say, "but if you write it as a block numeral, with straight lines, then the grapheme for three has three angles."

Really? So why didn't you write zero that way? Because when I last checked, a zero has four angles when written in block form.

And let's face it, even if you want to write your numbers in block form, no one writes them like this. Check your calculator. A two is an inverted five, which means that, like five, it has four angles. Four. Not five. Not two. Four.

And that ridiculous looking seven? Some people put a hook at the top of their sevens. Some people put a line through the middle. But do you know anyone who puts a line at the bottom of their sevens? I've been teaching math for a long time, and I don't think I've ever seen anyone write a seven with that serif. Oh, and I just checked through all 150 fonts I have installed on my computer, and none of them have that. It's just silly.

Do we really even need to talk about that absurd nine? Who in the world puts a curl on the curl on the curl at the bottom of their nines?

I get to the end of their numbers and find myself wondering, "Why? Why?? Why does this thing even exist?"

I have no idea. Seriously.

It's not even a valuable mnemonic for remembering anything. It's purposeless. And on top of that, it has zero angels. What's up with that? Why even mention angels, if you're not going to include any?

As one commenter said, "I never looked at the digits this way before. And now I'm never going to do it again."

Thanks, outlaw!

Lately I've been getting a spate of questions about meter in poetry. These question range from "Is this poem iambic pentameter?" to "How do you tell if a syllable is stressed or not?"

Let's start with the second question: How do you tell if a syllable is stressed or not?

If it's trembling uncontrollably, or hiding under the blankets, it's probably a stressed syllable.

Okay, serious answer. A stressed syllable is a syllable that has emphasis within a word (or within a line of poetry). So the best way to tell is to say the word in an overly dramatic way, choosing different syllables to emphasize. For example, let's say we have the word "emphasize," and we want to figure out which syllable is stressed. So we try saying it a few different ways. Try reading the following line, and SHOUT whenever you see a capitalized syllable (or, if you're in a library, whisper when you see a lower-case syllable).

EM-pha-size, em-PHA-size, em-pha-SIZE.

Clearly the middle one sounds WRONG! So we know that the middle syllable is unstressed. But the first and last ones, neither of them sounds horrible, but EM-pha-size definitely sounds better than em-pha-SIZE. EM is the stressed syllable in the word, and the other two are unstressed.  You could argue that SIZE has a secondary stress, but the general rule is, only one syllable in a word has the primary stress.

When I was a little kid I could never say the word "aluminum" properly. Let's see if we can work it out by syllables. Get ready to SHOUT again!

A-lu-min-um, a-LU-min-um, a-lu-MIN-um, a-lu-min-UM.

Clearly the one that sounds correct is the second one. Thus, LU is the stressed syllable.

The interesting thing is that if you put multiple words together, we may start hearing some of those "secondary" stresses more clearly:

EM-pha-SIZE a-LU-min-UM.

Those two words sound really great together, because we hear it as alternating stressed and unstressed all the way through. If we switch the order, though, it doesn't sound right to say it that way:

a-LU-min-UM EM-pha-SIZE.

Somehow, those two stressed syllables next to each other sound awkward and cumbersome. However, if we let the UM and SIZE get "swallowed up" and treat them as unstressed syllables:

a-LU-min-um EM-pha-size.

Now it sounds more relaxed and natural. Read this way, the two words sound natural together, because we have a stress every three syllables. It's important to remember that the context of words affects how they are stressed.

Another example of stress being affected by context is the word "present." This word will be pronounced pre-SENT if it is a verb, but PRE-sent if it is a noun. So you always have to think about the meanings of words to determine how they should be read.

So how do you tell if a poem is in a particular meter? Well, to answer that, you need to know what the meters are. For example, anapestic tetrameter means that you have two syllables unstressed, followed by a stressed syllable, and that is repeated four times in a line of poetry. So, for example, if you wanted to know if Robert Frost's "The Road not Taken" is anapestic, you could write it out with the stress on every third syllable:

two roads DI-verged in A yel-low WOOD.

Okay, that sounds just plain silly. I'd say Frost's poem is not anapestic.

What about "The Night Before Christmas"?

'twas the NIGHT be-fore CHRIST-mas and ALL through the HOUSE.

Hey! That sounds nice! That's the way it's supposed to be read! So yes, that is an anapestic poem.

Then there's iambic poetry. A poem is iambic if you start with an unstressed syllable, and then alternate stressed and unstressed. So let's try Shakespeare's famous line "It is the star to every wandering bark." We'll try writing that out with alternating stress:

it IS the STAR to EV-ery WAN-der-ING bark.

Uh oh...that doesn't sound quite right. It worked well until we got to the word "wandering." Then things went screwy. But you know, lots of times, when we say wandering, we don't actually pronounce that middle syllable; we say "wandring." So let's try it that way:

it IS the STAR to EV-ery WAN-dring BARK.

Again, this starts out sounding right, but about the time we get to the word "A," it goes south on us. Who wants to emphasize the word "a" in a poem? Well, that's the thing; Robert Frost knew that if people read his poem naturally, without thinking about meter, the words "in" and "a" would both kind of get swallowed up - almost as though they were one syllable:

Two ROADS di-VERGED (in a) YEL-low WOOD,
and SOR-(ry i) COULD not TRAV-el BOTH
and BE one TRAV'ler, LONG i STOOD
and LOOKED down ONE as FAR (as i) COULD
To WHERE it BENT (in the) UN-der-GROWTH.

Okay, now stop yelling, and go read that stanza in your natural voice. Isn't it beautiful? That's poetry for you!

Related post: How does context affect stress of syllables?