## A Multiplication Based Logic Puzzle

### Numbers up to 820 with Exactly 12 Factors

Let’s begin with today’s puzzle. Afterwards I’ll tell you a little about the number 820 and why I decided to make a list of all the numbers up to 820 with exactly 12 factors.

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

Now here’s some information about the number 820:

• 820 is a composite number.
• Prime factorization: 820 = 2 × 2 × 5 × 41, which can be written 820 = 2² × 5 × 41
• 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 x 2 x 2 = 12. Therefore 820 has exactly 12 factors.
• Factors of 820: 1, 2, 4, 5, 10, 20, 41, 82, 164, 205, 410, 820
• Factor pairs: 820 = 1 × 820, 2 × 410, 4 × 205, 5 × 164, 10 × 82, or 20 × 41
• Taking the factor pair with the largest square number factor, we get √820 = (√4)(√205) = 2√205 ≈ 28.635642

820 is the sum of two squares two ways:

• 28² + 6² = 820
• 26² + 12² = 820

Because 5 and 41 are two of its prime factors, 820 is the hypotenuse of FOUR Pythagorean triples:

• 180-800-820 which is 20 times 9-40-41
• 336-748-820 which is 4 times 84-187-205, but it could also be calculated from 2(28)(6), 28² – 6², 28² + 6²
• 492-656-820 which is 164 times 3-4-5
• 532-624-820 which is 4 times 133-156-205, but it could also be calculated from 26² – 12², 2(26)(12),26² + 12²

Since 820 = (40×41)/2, we know that 820 is the 40th triangular number, thus

• 1 + 2 + 3 + . . . + 38 + 39 + 40 = 820

820 is also a palindrome in four other bases:

• 1010101 BASE 3 because 3⁶+3⁴+3²+3⁰=820
• 1111 BASE 9 because 9³+9²+9¹+9⁰=820
• 868 BASE 11 because 8(11²)+6(11¹)+8(11⁰)=820
• 1I1 BASE 21 (I is 18 base 10) because 21²+18(21¹)+21⁰=820

Below is a chart of the numbers up to 820 with exactly 12 factors. Notice that two sets of consecutive numbers, (735, 736) and (819, 820), are on the list. Look at their prime factorizations:

• 735=3×5×7², 736=2⁵×23
• 819=3²×7×13, 820=2²×5×41

Those prime factorizations mean that while 735 and 736 are the smallest consecutive numbers with exactly 12 factors, 819 and 820 are the smallest consecutive numbers whose prime factorizations consist of one prime number squared and exactly two other primes. Thanks to Stetson.edu for alerting me to that fact. Here’s something interesting about the chart: of the 77 numbers listed, only six are odd numbers.

So, how did I know what numbers to put on the list?

In order to determine how many numbers up to 820 have exactly 12 factors, we must first factor 12. We know that 12=12, 6×2, 4×3, and 3×2×2.

Next we subtract 1 from each of those factors to determine the exponents we need to use:

12 gives us 12-1=11. For prime number a, when is a¹¹ not larger than 820? Never, because 2¹¹>820.

6×2 gives us 6-1=5 and 2-1=1. For prime numbers a and b, with a≠b, when is ab¹ less than or equal to 820? These nine times:

• 2×3=96, 2×5=160, 2×7=224, 2×11=352,
• 2×13=416, 2×17=544, 2×19=608, 2×23=736
• 3×2=486

4×3 gives us 4-1=3 and 3-1=2. For prime numbers a and b, with a≠b, when is a³b² not larger than 820? These six times:

• 2³×3²=72, 2³×5²=200, 2³×7²=392
• 3³×2²=108, 3³×5²=675
• 5³×2²=500

3×2×2 gives us 3-1=2, 2-1=1, and 2-1=1. For distinct prime numbers a, b, and c, when is a²bc not larger than 820? 52 times. Here’s the breakdown: It happens 35 times when 2²=4 is the square number:

And it happens another 27 times when a prime number other than 2 is squared:

That’s a lot of numbers with exactly 12 factors! After I sorted all the numbers that I found in numerical order, I was able to make that chart of numbers up to 820 with exactly 12 factors, and yes 819 and 820 are the smallest two consecutive numbers whose prime factorization consists exactly of one prime number squared and two other prime numbers.

### 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.

———————————

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.)

**********************************

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

**********************************

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

### 818 How Many Steps Do You Take Each Day?

My brother, Doug, recently visited me. He told me about his goal to get 11,000 steps every day. The American Heart Association recommends 10,000 steps a day.  Is it worth trying to get a thousand steps more than the recommended number?

My brother shared the cool mathematics of an 11,000 daily step goal with me, and now I want to share it with you:

Now I think 11,000 steps a day is a very worthy goal! It helps me see the big picture of 1,000,000 steps each quarter and 4,000,000 steps each year and that will help me be more likely to meet the 11,000 step goal EVERY day.

I wear a Fitbit to keep track of my steps everyday. A fellow blogger recently wrote a fun poem about wearing a Fitbit to keep track of steps, and it made me smile.

It will take you far fewer than 11,000 steps to complete this multiplication table puzzle. It isn’t the most difficult puzzle I make, but it can still be a challenge:

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

Now here is a little bit about the number 818:

• 818 is a composite number.
• Prime factorization: 818 = 2 x 409
• 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 818 has exactly 4 factors.
• Factors of 818: 1, 2, 409, 818
• Factor pairs: 818 = 1 x 818 or 2 x 409
• 818 has no square factors that allow its square root to be simplified. √818 ≈ 600699

818 looks the same right side up or upside down so we call it is a strobogrammatic number.

23² +  17² = 818

Finally, 818 can be found in these two Pythagorean triple equations:

• 240² + 782² = 818²
• 818² + 167280² = 167282²

### 817 Tiny Squares

Start at the top row of this level three puzzle and work down one cell at a time until you’ve written each number from 1 through 12 in both the top row and the first column. You will have solved the puzzle if all the clues given in the puzzle are the products of the numbers you wrote. You can do this!

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

• 817 is a composite number.
• Prime factorization: 817 = 19 x 43
• 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 817 has exactly 4 factors.
• Factors of 817: 1, 19, 43, 817
• Factor pairs: 817 = 1 x 817 or 19 x 43
• 817 has no square factors that allow its square root to be simplified. √817 ≈ 28.58321

817 is the sum of three consecutive prime numbers:

• 269 + 271 + 277 = 817

Because it has two factor pairs in which the numbers in the pair are either both odd or both even, 817 can be written as the difference of two squares two different ways:

• 817 = 31² – 12², which I’ve illustrated below.
• 817 = 409² – 408²

Being able to be written as the difference of two squares means that 817 is a leg in two Pythagorean triples so:

• 744² + 817² = 1105²
• 817² + 333744² = 333745²

Last, but certainly not least, 817 is the difference of two consecutive cubes, namely 17³ – 16³ = 817. That means that 817 is the 17th centered hexagonal number as well! There are 17 squares on each side of the figure below and 817 tiny purple squares in all. The horizontal row in the middle has 2(17) – 1 = 33 tiny purple squares.

2(17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32) + 33 = 817.

That difference of two cubes also means that

• 817 = 17³ – 16³ = (17 – 16)(17² + 16·17 + 16²) = (1)(289 + 272 + 256) = 817.

That’s from a³ – b³ = (a – b)(a² + ab + b²) and is illustrated below using 2D cubes instead of squares. In 3D, when the sides are folded up, the darker cubes can be claimed by two different sides so the figure will look like a 17 x 17 x 17 inside corner such as in a room where three perpendicular lines meet, but in 2D it’s just the flat net you see here:

### 816 and Level 2

Eight is half of sixteen, so 816 is divisible by 6. You probably weren’t expecting that divisibility rule, but it’s true.

816 can also be easily divided by 2, 4, and 8. How many factors does 816 have in all? Plenty! Scroll down past the puzzle and see!

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

• 816 is a composite number.
• Prime factorization: 816 = 2 x 2 x 2 x 2 x 3 x 17, which can be written 816 = 2⁴ x 3 x 17
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 816 has exactly 20 factors.
• Factors of 816: 1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 34, 48, 51, 68, 102, 136, 204, 272, 408, 816
• Factor pairs: 816 = 1 x 816, 2 x 408, 3 x 272, 4 x 204, 6 x 136, 8 x 102, 12 x 68, 16 x 51, 17 x 48 or 24 x 34
• Taking the factor pair with the largest square number factor, we get √816 = (√16)(√51) = 4√51 ≈ 28.5657

Since 17 is one of its factors, 816 is the hypotenuse of a Pythagorean triple:

• 384-720-816 which is 48 times 8-15-17

816 is repdigit OO in base 33 (O is 24 base 10). That is true because

• 24(33¹) + 24(33º) = 24(33¹ + 33º) = 24(33 + 1) = 24 × 34 = 816

816 is the sum of the sixteen prime numbers from 19 to 83:

• 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 816

Coincidentally, 816 is also the sixteenth tetrahedral number.

That’s because 16(16 + 1)(16 + 2)/6 = 816, which is a fast way to compute it. Here’s what it means to be the 16th tetrahedral number:

### 815 and Level 1

Since multi-digit 815 ends with 5, it is a composite number, and it is also the hypotenuse of a Pythagorean triple:

• 489-652-815 which is 163 times 3-4-5.

Can you write the numbers 1 – 12 in both the first column and the top row so that this puzzle functions like a multiplication table?

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

• 815 is a composite number.
• Prime factorization: 815 = 5 x 163
• 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 815 has exactly 4 factors.
• Factors of 815: 1, 5, 163, 815
• Factor pairs: 815 = 1 x 815 or 5 x 163
• 815 has no square factors that allow its square root to be simplified. √815 ≈ 5482048.

My NEWEST grandchild was adopted a couple of months ago in China. My daughter-in-law blogged about picking up their daughter and returning to her orphanage to say good-bye before they left China. The details given are very moving. In spite of the traumatic start, this little girl and her family have grown to love each other very much.

Here she is sitting with my husband, me, and her big sister. My husband and I are pretty new to her so she’s probably thinking in Mandarin, “Who are these people?” On the other hand, we are delighted to be a part of her life now.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors 807-814

Here’s a little about the number 813:

• 813 is a composite number.
• Prime factorization: 813 = 3 x 271
• 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 813 has exactly 4 factors.
• Factors of 813: 1, 3, 271, 813
• Factor pairs: 813 = 1 x 813 or 3 x 271
• 813 has no square factors that allow its square root to be simplified. √813 ≈ 51315.

813 is repdigit 111 in BASE 28 because 28² + 28¹ + 28º = 813

Stetson.edu and my calculator informed me that 813^e = 81,366,615.06223032 . . .