Games
Problems
Go Pro!

Ask Professor Puzzler


Do you have a question you would like to ask Professor Puzzler? Click here to ask your question!
Yearly archive for 2015.

Eleventh grader Tiara asks the following question:

"How do you factor -1 + p2?"

Well Tiara, I'm going to give you two answers to that question. The first one is the answer you're probably looking for. It's simple, it's straightforward, and it's uncomplicated.

Then I'll give you an answer that no one would ever use, because it's overcomplicated - like using a sledgehammer to kill a mosquito. But I'm going to give it to you anyway, just because being overcomplicated amuses me.

Fly Swatter Approach

Reverse the order of the terms. Now instead of -1 + p2, you have p2 - 1, which is a Difference of Squares. Therefore, it factors into:

(p - 1)(p + 1)

Sledgehammer Approach

My ridiculous sledgehammer approach requires that you have a rudimentary understanding of i, the imaginary unit. If you don't understand that, all you really need to know is that i is used to represent the square root of -1.

If you include the imaginary unit in your thinking, then you can rewrite your expression like this:

-1 + p2 = i2 - p2i2 

And now you've got a difference of squares, so it factors like this:

(i - pi)(i + pi)

Each of these factors can have an i factored out, giving us:

i2(1 - p)(1 + p)
-1(1 - p)(1 + p)

Distributing that -1 through the first factor gives us (-1 + p)(1 + p), and rearranging the terms of the first factor gives us:

(p - 1)(p + 1), which is the same as the answer we obtained with the fly swatter method.

Who would ever want to use the second approach? No one. Except ridiculous math teachers who think it's funny to take simple problems and make them absurdly complex.

Or bloggers who think their blog post would have been too short otherwise.

Professor Puzzler

If you find this article helpful, please share it to help make everyone a little warmer and a little safer this winter.

The "you can make a space heater out of candles and a clay pot" thing has been seen floating around facebook again. If you don't know what I'm talking about, here's a picture. Sorry about the big word "NO!" across the image, but we don't want to be responsible for someone finding this image on our site and thinking it's a good idea.

The text reads:

"You can heat an entire room with this Terra-cotta pot turned space heater. You need a clay pot, some large bricks, and some candles. A good thing to know in case the power goes out this winter."

So why do we say "NO!" to this? Well, the answer is three-fold. Here are the three answers:

Reason #1: Information Source

You found it on the internet, from an un-vetted, non-scientific source, and it involves fire.

That should be reason enough right there. Period.

Remember that if your house burns down because your friend shared this image and you decided to try it, you're not even going to have the satisfaction of being able to sue someone over it.

Reason #2: Magical Heat Multiplication

Can you heat an entire room with just 4 tealights? Sure. If the temperature outside is not much lower than what you want the indoor temperature to be, and your room is well insulated, and you're willing to wait a long time. If those conditions aren't met, you can't do it. If the temperature differential between indoors and outdoors is high, and your house isn't well insulated, heat will be lost to the outdoors faster than your candles can provide heat. And even if your room is well insulated, it's going to take a very long time for those candles to do the job.

Look at it this way. According to Wikipedia, a single tealight has an energy output of about 100 BTU/hr. A small space heater, capable of heating a small room, is 5000 BTU/hr. Thus, you actually would need fifty candles to heat the room as efficiently as a space heater, instead of just four. So let's suppose a space heater would take one hour to heat the room up. It'll take the candles over twelve hours to do the job (considering the burn-life of a typical tealight is well under twelve hours, we have a serious problem!). And remember that this is under the "ideal" conditions, where you're not losing heat to the outdoors more quickly than the candles can produce it. Since the image says, "A good thing to know in case the power goes out this winter," they're suggesting very non-ideal conditions.

"So what," you might think, "It's not just the tealights! It's also bricks and a clay pot!"

Yep, and the bricks and the clay pot do not add a single BTU to the arrangement. They do not, can not, will not magically multiply the amount of heat produced by the candles. What they do provide is a way to concentrate that heat. So even though the rest of the room is cold, if you sit right next to this little heater, and hover your hands over it, you'll warm up, and be fooled into thinking the whole room is warming up as much as your hands are.

But you really don't want to hover your hands over this little space heater. Why not? Because...

Reason #3: Flash Point of Candle Wax

Did you know that candle wax does burn if it gets hot enough? How hot is "hot enough?" Well, I found a few candle-making references that put the number in the 200 - 250 degree Celsius range. That's actually not all that hot, considering a candle flame, at its hottest point, is more than 1000 degrees Celsius. Of course, we're kept safe by the simple fact that all that heat energy has somewhere to go (up and out!) But what if we surround our candles with bricks and a clay pot, which have the effect of trapping a lot of that heat in a small area?

We have the potential of raising the temperature of the candle wax to its flash point. Then everything ignites, and your hands (which are hovering over your little space heater) suddenly get very very very warm (also known as "third degree burn").

How likely is this to happen? I don't know, but I'm not going to try to find out. If you ever see this image floating around the internet, check to see if there are any comments associated with it. If there are, you'll see plenty of comments from people who had hazardous encounters with this device.

And so, when all is said and done, perhaps it would be better to invest in a little more thermodynamics knowledge...

A couple more comments on this before I let you go:

  • Since it would require 50 candles to match your space heater, you might be tempted to build about 12 of these things (48 candles, 12 clay pots, etc) to see what happens. Don't do this. Because every single one of those candles is producing unsafe gasses as they burn. Just a handful of candles is not going to hurt you, but if you have 48 candles in a small room, and that room is unventilated, you may be creating more serious problems for yourself. (Why is the room unventilated? Because you're trying to heat it up, so you don't have a window or door open. Bad idea.) The only unvented space heater that's safe to use indoors is an electric one.
  • If you've decided to ignore everything in this post and try it anyway (please don't!), remember that burning wax is like burning oil - it floats on water, and therefore pouring water on your fireball is only going to spread it.

Staying Safe and Warm,
Professor Puzzler

Kameshwar, from Kakinada asks, "Can we say 'The Himalayas are located to the north of India?' (use of definite article) and where it should be omitted?"

Thanks for asking, Kameshwar. I wish I could give you a definitive answer to this question, because I love mountains; hills, valleys, streams, lakes, oceans - they're all wonderful - but give me a mountain any day and I'm happy.

I don't actually know for sure the answer to your question. However, I did enough reading and research that I think I can give you a fairly reasonable rule of thumb.

The first stops in my research were the Encyclopaedia Britannica website and Wikipedia. Both used the name "The Himalayas" extensively throughout their articles, so I think it's safe to say that whenever you are talking about the mountain range as an object (noun) you can freely use the definite article "the," and no one will complain.

Britannica did omit the definite article in their page title; the title was "Himalayas," rather than "The Himalayas." However, I'm not sure I'd read anything into that; encyclopedias are designed to be read and searched alphabetically, so an article of any sort would interfere with alphabetizing. You should probably follow the example they set in the text of the article, and use "the."

Now, in the course of my reading, I did learn something new (which does relate to your question, and I'll get back to that in a minute!); "Himalayas" is an anglicized version of the word "Himalaya," which means (roughly) "the dwelling place of snow."

hima = snow
alaya = abode

How did we get from the local name for the mountains to our current version? Well, English speaking people thought, "If one mountain is a himalaya, then a bunch of them would be himalayas." However, there are two problems with this reasoning:

  1. The entire range is the "abode of snow," so you could argue that the range is a singular abode, not plural.
  2. Even if you did want to pluralize "himalaya," the proper Sanskrit pluralization of it is - are you ready for this? Himalaya.

That's right; we probably shouldn't be saying "himalayas" at all; we should be saying "himalaya." There is no distinction between the singular and plural of the word.

There is a lot of literature (mostly older) that uses "Himalaya" instead of "Himalayas," but it looks like our Anglicization of the word is here to stay. Since I have a friend who visited Nepal a few years back, I thought, "I should ask him how the locals referred to the mountains - whether they said himalaya or himalayas." Here's his response:

"In the fall of 2007 I spent 7 weeks Nepal...I never recall it being described as the Himalaya only the Himalayas."

He goes on to say that he can't specifically recall a local saying it one way or the other, so we can't take his account as gospel. My gut feeling is that if he'd heard someone calling them "Himalaya," he would have noted and remembered it. So perhaps even the locals have given in to the Anglicization of the name - at least in the presence of outsiders!

Okay, what does this Himalayas/Himalaya stuff have to do with your question? 

  • It seems to be more acceptable to drop the definite article if you say "Himalaya" instead of "Himalayas." Example: My friend visited the Himalayas in 2007. vs. My friend visited Himalaya in 2007. To English speakers, the second sentence might sound odd, but our English-trained minds quickly interpret "Himalaya" as a singular region in which the plural "himalayas" reside, so it works out comfortably.
  • Did you know that the word "himalaya" also functions as an adjective? You can go to a himalaya spa, or a himalaya restaurant (where you would experience himalaya cuisine). You can also use "himalayas" and "himalayan" as adjectives, but neither flows off the tongue as comfortably; there's an elegance to the sound of the word without any closing consonant. But regardless of whether you use himalaya, himalayas, or himalayan, if it's an adjective, you do not use a definite article with it.

So, after all of that, my final summary of the answer is: when using "Himalayas" as a noun, by all means, put "the" in front of it. But if you're using it as an adjective, don't!

Thanks for asking the question; this was an interesting one to research, and seeing all those beautiful pictures of majestic mountains makes me want to visit Himalaya!

Professor Puzzler

 

Ninth grader Racheal from Nigeria asks the following question: "How Do I Analyze a Poem?"

This is a very general question, and the degree of specificity with which you analyze poetry will depend a great deal on your educational level, when the poem was written (and by whom), and what your teacher/professor expects, if you are doing it for a school assignment. But here are some of the most common aspects of poetry that you can look at when analyzing a poem. Much of the information provided here we have previously written up on our sister site Fifteen Minutes of Fiction, and so in some cases we will simply link to short articles where you can read more.

Form

When you first look at a poem on a page, its general form is likely the first aspect that you will notice. Is it long, or short? Are the lines mostly uniform in length, or do they vary? Are the stanzas all the same length, or are they different?

The form has to do with the way the poem is organized. There are poets who simply let their words flow in paragraph form in a sort of “stream of consciousness,” but most will construct the line breaks in specific ways so as to emphasize rhyme and meter, or influence the way the reader reads the poem.

A poem is made up of lines, which are usually organized into groupings known as stanzas. Stanzas come in different sizes, from just two together (known as a couplet) to many. Poems are often written to follow a specific form, such as a sonnet or haiku. Sonnets are fourteen lines of iambic pentameter with a specific rhyme scheme (more on that in a moment). Haikus are very short, consisting of only three lines, with five syllables in the first and last lines, and seven in the middle line. Some other popular forms include villanelle (a form that involves repeated lines in a specific pattern), terza rima (a form of three-line stanzas with a specific pattern) and blank verse (a form that has a specific meter to it, but no rhyme scheme).

As you become familiar with some of these forms, you will be more likely to recognize them when you see them. This will be one of the first things to draw your attention when you see a poem for the first time.

Meter

The meter of a poem has to do with the way the syllables of words are arranged to produce a pattern of emphasis and non-emphasis. Syllables that are emphasized are called “stressed” syllables, while those that are not are called “unstressed.”

One of the most common meters is iambic pentameter. An iamb is an unstressed syllable followed by a stressed syllable. Five of those together make a line of iambic pentameter.

If you reversed the order and started with one stressed syllable, and followed it with an unstressed syllable, the result would be a trochee. A line of trochaic pentameter would be the same length as a line of iambic pentameter, but the alternating stressed and unstressed syllables would be in the opposite order.

There are also two fairly common meters that are based on a pattern of three syllables instead of two. These are anapests (unstressed, unstressed, stressed), and dactyls (stressed, stressed, unstressed).

Not all poems have meter. Poetry that is referred to as “Free Verse” does not have meter, though it may still incorporate rhyme.

For more information about meter, as well as poetic examples, follow the links I’ve given in the text.

Rhyme Scheme

Most poems with a meter also incorporate rhyming at the end of each line, though blank verse is an exception. Poems do not have to rhyme, but when they rhyme at the end of each line, there is often a pattern to the rhymes.

Elizabethan Sonnets, for example, follow an ABAB rhyming pattern. This means that ends of the first and third lines rhyme with each other, and the second and fourth do the same.

In some poems, quatrains follow an ABCB pattern instead, where only the ends of the second and fourth lines rhyme with each other.

Terza Rima poems use an interesting rhyme scheme, where each stanza has three lines, and the first and last rhyme, while the middle line rhymes with the first and last lines of the next stanza, so the pattern goes: ABA, BCB, CDC, DED, etc.

Content

So far, we’ve talked about things we see and hear about poems: their forms on the page, their rhyming patterns, and metrical beats. But beyond form, we can also analyze poems by looking deeper at the meaning of the words. This can be more difficult. It’s one thing to identify the particular patterns a poet has utilized in bringing their words to life, but how do those structures serve the words that they provide support for, and what are those words saying?

There are many other things you can identify within the text of a poem: use of simile and metaphor, personification. Identifying these things can help you to get a better idea of what the poem is talking about.

When a poem makes observations, the observations are coming from the speaker of the poem. The speaker may very well be the author too, but sometimes authors write from the perspective of someone or something else. That’s something to keep in mind when we’re discussing poetry—we don’t say “Here the author is saying that he doesn’t like winter,” when, even if the poem seems quite vehement against winter, it may simply be that the author is writing from the perspective of someone else, even if the word “I” is used. So we say, “Here the speaker seems to be saying…” Poems can be a wonderful tool of fiction, and so we shouldn’t limit analysis to simply what we think the real-life author is saying.

There are many more examples I could give, but I’ll leave you with a few questions you can ask when reading a poem, that may help you come to a better understanding of the content.

When was this poem written, and by whom?

Who is the “speaker” of this poem?

Does this poem have an identifiable subject—something that it primarily describes or discusses?

Which senses (if any) is the speaker drawing on in their poetic descriptions (sight, sound, touch, smell, taste)? (Sometimes it can be useful to go through the poem and highlight all the adjectives)

Does the speaker use any similes or metaphors? Are they comparing one thing to another?

Does this poem strike you as a fictitious account, or is it something that could really happen?

Are there any significant repeated words or phrases?

Pegi writes: "I recently switched from http to https because it seemed to my IT guy that the reason we could not collect data on Google Analytics from a site that was https, was because we lacked the magic S....so he got the certification  done...

BUT......now I see that ALL the previous links people had to my  site, are now going to be lost because of the switch...

Can I have both address....and drive the old http to my new https ?

And, are you a Google Analytics expert?

Thank you."

Well, Pegi, first let's talk about your big (unspoken) question - do you need https in order to collect Google Analytics data?

No, you do not. In fact, I'm using Google Analytics on this site, and you'll notice that there's no https in the address for the page you're looking at. So your IT guy may have set up a security certificate for nothing.

Let's talk for a moment about why you might want to have an SSL certificate, even though it probably won't make any difference to Google Analytics.

If you sell anything on your site, you'll want a certificate, because payment processing must occur on an encrypted page. If you're selling something, but using an outside service (like PayPal) to process payments, the security certificate might not be necessary, but people will feel safer if it's there. I know that when I start creating a shopping cart on a site, one of the first things I do is look for that "magic S." If it's not there, I never get far enough in the process to find out whether they're processing my payment information elsewhere.

Now let's talk about why you might not want SSL on your site. The big reason is that every page viewed over SSL is encrypted, and the encryption requires more information transferred over the internet connection. Why does this matter? Well, this will matter for two reasons:

  1. If you are limited on bandwidth (which is probably not likely - most hosting companies are offering enough bandwidth that easily cover the needs of most small businesses) then you're using more bandwidth every time someone views a page.
  2. Because the pages are encrypted, they are taking more time to load. You might have noticed that your page loads a little slower now that it's on a secure socket. On the other hand, you might not have noticed; the big difference will be noticed primarily by those who are not on high speed internet. And they are a vanishing breed.

In addition, of course, you have to pay for it every year. In the grand scheme of things it's not a big expense, but it is an expense.

So what are your options now?

  • Keep SSL, and keep it on the whole site. Your concern that http links will be lost is really not something to be concerned about for you, because if someone clicks a link to an http page, your server automatically redirects them to the https version of the same page. (I know this, because I visited your site, and did so without the https.)
  • Keep SSL, but use it only on parts of your site. As I mentioned before, the page you reading right now isn't on a secure connection, but we do have a security certificate on this site - go to the site's Shopping Page, and you'll see the "magic S". If you try to visit the shopping page with http, you get redirected to the https version. Similiarly, if you try to visit any other page of the site with https, you get redirected to the page without it.
  • Get rid of the SSL. This may be problematic, if you've had the certificate for awhile, because if you have a security certificate, both http and https links will work. But if you get rid of https, then https links will no longer work, and anyone who has an https page bookmarked, or finds your site through Google (google has indexed your pages as https) will get a browser warning instead of your site. If you feel like you really need to get rid of the SSL at this point, that's probably a whole other blog post to explore the best way to handle that without losing much traffic.

Your final question - am I a Google Analytics expert? No, I'm really not. I use it on all my sites, though, so if you had additional questions, I might be able to help out.

I hope this was helpful!

Regards,
Professor Puzzler

 

Older posts

Understanding Coronavirus Spread

A Question and Answer session with Professor Puzzler about the math behind infection spread.

Blogs on This Site

Reviews and book lists - books we love!
The site administrator fields questions from visitors.
Like us on Facebook to get updates about new resources
Home
Pro Membership
About
Privacy