## A Multiplication Based Logic Puzzle

### 819 How to Type eˣᵖᵒⁿᵉⁿᵗˢ in WordPress

Exponents, ⁰¹²³⁴⁵⁶⁷⁸⁹, are written to the right of their base numbers and a little higher. They are about half the height and about half the width of the base number, too.

Exponents are important to me. They and other special characters allow me to include factoring information and interesting number facts on every post I write. For example …

• 819 is a composite number.
• Prime factorization: 819 = 3 × 3 × 7 × 13, which can be written 819 = 3² × 7 × 13
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 819 has exactly 12 factors.
• Factors of 819: 1, 3, 7, 9, 13, 21, 39, 63, 91, 117, 273, 819
• Factor pairs: 819 = 1 × 819, 3 × 273, 7 × 117, 9 × 91, 13 × 63, or 21 × 39
• Taking the factor pair with the largest square number factor, we get √819 = (√9)(√91) = 3√91 ≈ 28.618176

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² = 819, making 819 the 13th square pyramidal number.

315² + 756² = 819² so 819 is the hypotenuse of a Pythagorean triple.

2⁹ + 2⁸ + 2⁵ +  2⁴ + 2¹ + 2⁰  = 819 because 819 is 1100110011 in BASE 2.

I like that pattern of 1’s and 0’s. Here’s a few more of 819’s cool number patterns:

• 3⁶ + 3⁴ + 3² = 819 because 819 is 101010 in BASE 3.
• 3·4⁴ + 3·4² + 3·4⁰ = 819 because 819 is 30303 in BASE 4.
• 3·16² + 3·16¹ + 3·16º = 819 because 819 is 333 in BASE 16.

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Okay. Enough about 819. HOW do we type exponents when we write a blog?

Option #1: WordPress gives us some special characters in the editor. I’ve put red boxes around the exponents so you can find them faster:

As you can see, the WordPress’s editor only offers us º ¹ ² ³ ª as exponents, and they MIGHT fill all your needs. (Who am I kidding?) You can get to any of the symbols shown above by clicking on the Ω symbol in YOUR WordPress editor. I’ve put a red box around the Ω special character symbol in the PICTURE of the editor below.

Those symbols are good if you’re writing x³ or even 8¹º³². But what if you want to write an expression with a 4, 5, 6, 7, 8, or 9 as part of the exponent? Do you really have to settle for (2^7)×(3^5) when you really want to type 2⁷×3⁵? That carrot ^ symbol can look needlessly intimidating to people even if they are familiar with exponents.

So how do we type all those other exponents in WordPress? That is something I have been frustrated about and have googled about many times. I’ve read about and tried a couple more options: Superscripts and LaTeX.

Option #2 Superscripts: When I followed the superscripts’ directions for WordPress, and typed e<sup>xponents</sup> in the text editor as instructed, it made beautiful eˣᵖᵒⁿᵉⁿᵗˢ in the visual editor, but look at all these exponents marked in red, they fell down when I published this post. That is not acceptable. Other people may be able to get those superscripts to stay up, but I have tried repeatedly without success for over a year. On a related note: While writing this post I learned something useful about Microsoft Word. If you push down the Shift, Ctrl, and = keys at the same time, you can type in superscript in a Microsoft Word document. (You press the same keys to get out of superscript mode). Unfortunately, if you copy and paste that superscript writing into WordPress, it will look like superscripts in the visual editor, but not in your published work.

Option #3. LaTeX can be a great looking option. Still, when the exponents from the WordPress editor are typed alongside those in LaTeX, they can look a little wobbly: 2³ + $2^4$ + $2^5$ + 7² + $3^4$. In addition, LaTeX looked like LaTeX notation instead of exponents when I tried to use it in the title of this post.  The biggest drawback: LaTeX looks good when it’s published, but it is practically unreadable when it’s being typed. For example, without spaces, you must type [ latex ]2^4[ /latex ], just to get 2⁴. (If I took out the spaces it would read 2⁴ instead of showing you what LaTeX notation looks like.)

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This week I found a good 4th option: Microsoft Word has quite a few exponents, and WordPress liked them!

I’ve gathered the superscripts of the English alphabet and numbers from Microsoft Word in one place and included them here for the convenience of all other bloggers, making this post a great 5th option. Copy what you need from here, or copy and paste the whole list into a handy document of your own. True, not every letter of the English alphabet is available as an exponent in Word, but most of them are. This is the method I used to include eˣᵖᵒⁿᵉⁿᵗˢ in the title of this post.

x⁰¹²³⁴⁵⁶⁷⁸⁹ᴬᵃᵅᴮᵇᶜᴰᵈᴱᵉᶠᴳᵍᴴʰᴵⁱᴶʲᴷᵏᴸᴹᵐᴺⁿᴼᵒᴾᵖᴿʳˢᵀᵗᵁᵘⱽᵛᵂʷˣʸᶻ ⁺ ⁻ ⁼ ⁽  ⁾ Those superscripts or exponents stay up! And…these subscripts stay down! ₉₈₇₆₅₄₃₂₁₀ ₊ ₋ ₌ ₍ ₎ aₐ eₑ  jⱼ oₒ xₓ. Curiously,  hₕ kₖ lₗ mₘ nₙ pₚ sₛ tₜ seem to stay down on home computers but disappear on smart phones so you might not want to use them.

Subscripts are often used in notation for Permutations like ₆P₃ or Combinations like ₆C₃. (Those links will take you to some useful online calculators.)  Subscripts used with “⅟ ” can write infinitely many unit fractions like ⅟₃₂₁. Subscripts can also be used to write the base of logarithms such as log₂4=2.

Here’s a bonus, the Greek letters: Some of the Greek letters have superscripts and/or subscripts next to them, while others do not. For some reason unknown to me, Microsoft Word didn’t give π either one. (I could not have written this part of the post without zooming to 175% first. You might want to do that before using any of these, too.):

Ααᵅ, Ββᵝᵦ, Γ⸀γᵞᵧ, Δδᵟ, Εεᵋ, Ζᶻζ, Ηη, Θᶱθᶿ, Ιᶦιᶥ, Κκ, Λᶺλ, Μμ, Νᶰν, Ξξ, Οο, Ππ, Ρρ, Σσ, Ττ, Υυᶹ, Φᶲφᵠᵩ, Χχᵡᵪ, Ψψ, Ωω

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Those exponents from Microsoft Word will allow you to write important identities like the following without using awkward LaTeX notation:

• sin t = (eⁱᵗ – e־ⁱᵗ)/2i
• cos t = (eⁱᵗ + e־ⁱᵗ)/2

Back to the 4th option, Microsoft Word does include some other incomplete alphabets from other languages that are not included in my lists above. Here’s what you’ll need to do to get subscripts or superscripts from Microsoft Word:

1. In Microsoft Word click on “insert”,
2. click on “symbol”,
3. click on “symbol” (NOT “equation” because WordPress won’t copy anything you type there),
4. click on “more symbols”.
5. Next LOOK for the desired superscript or subscript on the chart. You may have to look for a while. Some of the them are listed together, while others seem to be randomly placed by themselves. For the alphabet, only use a letter that is in the top CENTER of its box. If you use a letter that is in the top LEFT of its box, you might end up typing something like 3 ͩͪ  or 7ͪͫ.
6. Type your expression in Word, then copy and paste it onto your blog.

So now you have been saved countless hours of frustration trying to type a few simple exponents or subscripts. Perhaps, now you can chance getting frustrated trying to solve this Level 5 puzzle?!

Print the puzzles or type the solution on this excel file: 12 factors 815-820

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### 411 Some Things You Should Know Before You Start a Blog

You’re passionate about some subject and are considering starting a blog. Here’s some things you should know first:

What your blog is about will eventually change to one degree or another. I have noticed many bloggers redefine themselves, some more than others. My blog was initially a way to share a factoring puzzle I invented, but most of the people who visit my blog are not looking for puzzles; they want to find the factors of a particular number. After all, my blog name implies that factors can be found here. To satisfy these viewers, I now give the factoring information for the number 411:

• 411 is a composite number.
• Prime factorization: 411 = 3 x 137
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 411 has exactly 4 factors.
• Factors of 411: 1, 3, 137, 411
• Factor pairs: 411 = 1 x 411 or 3 x 137
• 411 has no square factors that allow its square root to be simplified. √411 ≈ 20.2731

When I first noticed that people were visiting my blog because they wanted to know the factors of a number, I resisted changing, but later I gave in. I even edited all my previous blog posts to include factored numbers. Eventually I added the square roots of numbers and indicated if those square roots could be simplified. Still I keep my original intent and publish the puzzles I create.

Coming up with imaginative titles for EVERY blog post can be tedious or even impossible. Many people judge whether or not a post is worth reading by the title. Sometimes a great title pops into my head, but most of the time I’ve got nothing. I started giving my puzzle posts boring titles like “Level 3”  and actually had several posts with that same name. When my stats page told me that someone liked or visited a post called Level 2, I wasn’t sure which post that was. When I started factoring numbers, it solved my problem of giving each post a distinctive title even if it was still a boring name. Now I often use unimaginative titles like “406 and Level 4”.

Blogging uses up more time than you imagine it will. I once read a very long, hilariously funny, grammatically perfect post that the author claimed he wrote during his lunch time. I find that claim very hard to believe because it certainly doesn’t work that way for me. I can spend 15-30 minutes reading or writing blogs, but the clock at my house indicates that I actually spent two hours or more. No matter what clock I use, it’s the same story.

If you write a blog, you will connect with a community of people who read your blog and you read theirs. All that reading takes time. Commenting on someone else’s blog or answering a comment on your own blog also takes time. It may seem like you’re only spending a few minutes at a time, but it really is much longer than that. Family members may complain about other things not getting done or quality time not being spent even if they are trying to be supportive. You will find blogs you want to read even though its writer doesn’t feel the same way about your blog. You will learn so much reading other people’s blogs, and you may want to read more books about a particular subject or even write about it yourself. You will find yourself wishing there were more hours in the day, or that you could actually make some money blogging and cut back on your hours at your regular job.

Most bloggers don’t make money blogging. If you want to make money on WordPress, you have to have enough views, you have to purchase your own domain name, and you have to sign up for WordAds. After some consideration, I took this step in August 2014. I registered my domain name for \$18 a year and signed up for WordAds. I also pay another \$8 each year to keep my contact information private.

How many views do you have to have to make money on WordPress? Update: Since writing this post, WordPress allows ALL of its bloggers to sign up for Wordads. Now every blog with a registered domain name is a potential money maker, even if it turns out to be just a little bit of money each month. The amount of money that you’ll make will depend on how many views you have and where those views come from.

Was it worth it to register your own domain name? Yes, the number of views I’ve received after registering is much, much greater than it was before I registered. It is much easier to tell friends and acquaintances to look for Findthefactors.com than it was to tell them that name with wordpress added to it. Even though it cost more money than I might have made back, it helps me reach more viewers, and that helps me meet my original objective. The only drawback is that if don’t renew my registration every year Findthefactors.com could become what they call a parking lot. I would have no control over what people would see if they went to the site. If you read the fine print when you register your domain, this stipulation is made quite clear. Somebody else could put whatever they want on the unregistered site, including adult content. I intend to continue registering Findthefactor.com for the rest of my life. Hopefully my children and grandchildren will also after I’m gone.

If you’re still thinking of starting a blog, do it because you love a particular subject. Do it because it pleases you. There will be days when absolutely nobody else looks at your blog. You will gain some followers, but some of those followers will get discouraged with their statistics and quit blogging and quit reading your blog. You will also get some followers who follow you for the sole purpose of trying to get you to follow them back. Once you follow them, they never look at your blog again. If you blog, you have to do it for yourself, and if anyone else enjoys it, that is a bonus.

I chose WordPress because it allows me to put excel documents on my blog. (Blogspot didn’t.) Unfortunately people who view my blog on their smart phones can’t see those documents. I hope there are a few teachers out there who at least occasionally look at my blog on their home or school computers and print out the puzzles for their classes. That would make my decision to add those documents definitely worth it.

Fellow bloggers, what other things do you think new bloggers should know?

### 11 Counting Blessings

• 11 is a prime number.
• Prime factorization: 11 is prime.
• The exponent of prime number 11 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 11 has exactly 2 factors.
• Factors of 11: 1, 11
• Factor pairs: 11 = 1 x 11
• 11 has no square factors that allow its square root to be simplified. √11 ≈ 3.31662.

How do we know that 11 is a prime number? If 11 were not a prime number, then it would be divisible by at least one prime number less than or equal to √11 ≈ 3.3. Since 11 cannot be divided evenly by 2 or 3, we know that 11 is a prime number.

Sometimes 11 is a clue in the FIND THE FACTORS 1 – 12 puzzles, and the factors are always 1 and 11.

I have more blessings than I could ever completely count. This is not the place where I will attempt to name them one by one, but I wonder: is the number of blessings that I or anyone else has finite or infinite? Even being able to ponder that question is a blessing. In the last few years in the United States, much of the gratitude part of Thanksgiving has gotten lost in commercialism. Therefore, for some people the number of blessings may be finite and easily measured by counting things. Some of those blessings may be more imaginary than real. Nevertheless, there are still people who can see the hand of God all around them. For them the number of blessings is infinite. Likewise those who rely on the Savior and His infinite atonement have an infinite number of blessings. As I count blessings, I find that some of them are prime, and some are a composite of several blessings working together. Some blessings are rather odd while others are shared evenly. I am grateful for many positive events in my life, but even negative experiences are blessings because they have helped me to grow.

The following blessings may seem trivial, but I am grateful that WordPress has given me a way to share the Find the Factors puzzles not only as jpg pictures, but also in an excel file.  The puzzles have been a blessing to me, and I want to show my gratitude by sharing them with other people. I am grateful for the blogs I follow. They challenge me, entertain me, and teach me so much. I am also thankful to everyone who has looked at my blog.

Click 12 Factors 2013-11-28 to see the same puzzles in excel.