A Multiplication Based Logic Puzzle

Archive for the ‘Level 5 Puzzle’ Category

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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813 My Newest Grandchild

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 . . .

 

804 Is There a Pot of Gold at the End of This Rainbow?

Factor Rainbows can be a wonderful way to display the factors of a number. Not only are all the factors listed in order from smallest to greatest, but the factor pairs are joined together with the same color band.

The number 804 has 12 factors so it makes a lovely rainbow with 6 different color bands.

Is there a pot of gold at the end of this factor rainbow? I’ll let you decide the answer to that question.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Finding golden nuggets of information about a number might be less difficult than finding pots of gold.

I always begin the painstaking mining process by looking at the factors of the number:

  • 804 is a composite number.
  • Prime factorization: 804 = 2 x 2 x 3 x 67, which can be written 804 = (2^2) x 3 x 67
  • 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 804 has exactly 12 factors.
  • Factors of 804: 1, 2, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804
  • Factor pairs: 804 = 1 x 804, 2 x 402, 3 x 268, 4 x 201, 6 x 134, or 12 x 67
  • Taking the factor pair with the largest square number factor, we get √804 = (√4)(√201) = 2√201 ≈ 28.3548937575

About half of everything there was already in the factor rainbow.

Finding nuggets of information about the number 804 has been a little difficult and disappointing:

  • None of 804’s prime factors can be written as 4N+1, so 804 is NOT the hypotenuse of any Pythagorean triples.
  • 804 is NOT a palindrome in base 36 or any base less than that.
  • 804 is NOT the sum of any consecutive prime numbers.

Even though I did not find any golden nuggets in those places, I kept looking and finally found a couple of gems about the number 804:

804 can be written as the sum of three squares four different ways, and all of those ways have some definition of double in them:

  • 28² + 4² + 2² = 804
  • 26² + 8² + 8² = 804
  • 22² + 16² + 8² = 804
  • 20² + 20² + 2² = 804

Stetson.edu also gives us a nugget about the number 804 that may be a bit too heavy for most people to handle: “804 is a value of n for which 2φ(n) = φ(n+1).” That basically means that there are exactly half as many numbers less than 804 that are NOT divisible by its prime factors (2, 3, or 67) as there are numbers less than 805 that are NOT divisible by its prime factors (5, 7, or 23).

I started looking for golden specs about 804 in places that I don’t usually look.

267 + 268 + 269 = 804 so 804 is the sum of 3 consecutive numbers.

As stated before 804 is never the hypotenuse of a Pythagorean triple. However to find all the times it is a leg in a triple will require a lot of labor especially since 804 has so many factors, including 4, and two of its factor pairs have factors where both factors are even.

  • 134 × 6 is an even factor pair, so (134 + 6)/2 = 70, and (134-6)/2 = 64. Thus 804 = 134·6 = (70 + 64)(70 – 64) = 70² – 64² .
  • 402 × 2 is another even factor pair, so (402 + 2)/2 = 202, and (402 – 2)/2 = 200. Thus 804 = 402·2 = (200 + 2)(200 – 2) = 202² – 200²
  • Likewise odd or even sets of factor pairs of any of 804’s factors can also be used to find Pythagorean triples.

So to find all Pythagorean triples that contain the number 804, we will have to find all the times 804 satisfies one of these FOUR conditions:

  1. 804 = 2k(a)(b) so that 804 is in the triple 2k(a)(b), k(a² – b²), k(a + b²) OR the triple k(a² – b²), 2k(a)(b), k(a + b²).
  2. 804 = 2(a)(b) so that 804 is in the triple 2(a)(b), a² – b², a + b² OR the triple a² – b², 2(a)(b), a + b².
  3. 804 = a² – b² so that 804 is in the triple a² – b², 2(a)(b), a + b² OR the triple 2(a)(b), a² – b², a + b².
  4. 804 = k(a² – b²) so that 804 is in the triple k(a² – b²), 2k(a)(b), k(a + b²) OR the triple 2k(a)(b), k(a² – b²), k(a + b²).

Let the mining process begin! I’ll list the triples with the shortest legs first and color code each triple according to the condition I used.

  • 335-804-871 which used 804 = 2·67(3)(2) to make a triple that is 5-12-13 times 67
  • 603-804-1005 which used 804 = 2·201(2)(1) to make a triple that is 3-4-5 times 201
  • 804-1072-1340 which used 804 = 268(2² – 1²) to make a triple that  is 3-4-5 times 268
  • 804-2345-2479 which used 804 = 2·67(6)(1) to make a triple that is 12-35-37 times 67
  • 804-4453-4525, which used 804 = 2(6)(67)
  • 804-8960-8996, which used 804 = 70² – 64² or 804 = 4(35² – 32²) to make a triple that is 201-2240-2249 times
  • 804-17947-17965, which used 804 = 2(134)(3)
  • 804-26928-26940 which used 804 = 12(34² – 33²) to make a triple that is 67-2244-2245 times 12
  • 804-40397-40405, which used 804 = 2(201)(2)
  • 804-53865-53871 which used 804 = 2·3(134)(1) to make a triple that is 268-17955-17957 times 3
  • 804-80800-80804 which used 804 = 202² – 200²  or 804 = 4(101² – 100²) to make a triple that is 201-20200-20201 times 4
  • 804-161603-161605, a primitive Pythagorean triple, that used 804 = 2(402)(1)

If you look for a pot of gold at the end of a rainbow, you’re bound to be disappointed. Science/How Stuff Works just had to crush dreams and dispel 10 Myths About Rainbows. Unfortunately a pot of gold being at the rainbow’s end is included on that list. Still I suppose we could still put every golden spec or nugget about 804 into a little pot and call it a pot of gold.

Or if you are as clever and quick as a leprechaun, perhaps you will consider finding Pythagorean triples to be like finding pots of gold.

798 Cupid’s Arrow and Target

Here are two puzzles that go together and yet look out of sync. Sometimes cupid’s arrow reaches its target, and sometimes it doesn’t.

798-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-795-799

Here’s a little about the number 798:

798 is made from three consecutive numbers (7, 8, and 9), so it is divisible by three. The middle number, 8, is not divisible by three, so 798 is NOT divisible by nine.

798 is a palindrome in two bases:

  • 666 BASE 11 because 6(121) + 6(11) + 6(1) = 798
  • 383 BASE 15 because 3(225) + 8(15) + 3(1) = 798

798 is also the sum of two consecutive prime numbers: 397 + 401 = 798.

798 can be written as the sum of three squares four different ways:

  • 26² + 11² + 1² = 798
  • 25² + 13² + 2² =798
  • 23² + 13² + 10² = 798
  • 22² + 17² + 5² = 798

Here is 798’s factoring information:

  • 798 is a composite number.
  • Prime factorization: 798 = 2 x 3 x 7 x 19
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 798 has exactly 16 factors.
  • Factors of 798: 1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798
  • Factor pairs: 798 = 1 x 798, 2 x 399, 3 x 266, 6 x 133, 7 x 114, 14 x 57, 19 x 42, or 21 x 38
  • 798 has no square factors that allow its square root to be simplified. √798 ≈ 28.24889.

798-factor-pairs

793 and Level 5

793 is the sum of two squares TWO different ways!

  • 28² + 3² = 793
  • 27² + 8² = 793

Notice that 793 is 4(198) + 1, and neither 28 and 3 or 27 and 8 have any common prime factors. Could 793 possibly be a prime number?

The answer is no for two reasons:

  1. √793 is about 28.2, so we only need to check to see if 793 is divisible by 5, 13, or 17 (all the 4N+1 prime numbers that are less than 28.) However, 793 ÷ 13 = 61, so  793 isn’t prime.
  2. Also, any number that can be written as the sum of two squares in more than one way is never a prime number.

Both 13 and 61 give a remainder of one when they are divided by four, and 793 is the hypotenuse of FOUR Pythagorean triples:

  • 143-780-793, which is 13 times 11-60-61
  • 168-775-793, a primitive calculated from 2(28)(3), 28² – 3², 28² + 3²
  • 305-732-793, which is 61 times 5-12-13
  • 432-665-793, a primitive calculated from 2(27)(8), 27² – 8², 27² + 8²

793 is also palindrome 191 in BASE 24; note that 1(24²) + 9(24) + 1(1) = 793

Finally Stetson.edu informs us that 793 is 2(397) – 1.

793-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 793 is a composite number.
  • Prime factorization: 793 = 13 x 61
  • 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 793 has exactly 4 factors.
  • Factors of 793: 1, 13, 61, 793
  • Factor pairs: 793 = 1 x 793 or 13 x 61
  • 793 has no square factors that allow its square root to be simplified. √793 ≈ 28.16025568.

793-factor-pairs

786 and Level 5

786 is even so it is divisible by 2. Also since 786 is made from 3 consecutive numbers, we can tell automatically that it is divisible by 3. Those two facts together mean 786 is also divisible by 6.

  • 786 is a composite number.
  • Prime factorization: 786 = 2 x 3 x 131
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 786 has exactly 8 factors.
  • Factors of 786: 1, 2, 3, 6, 131, 262, 393, 786
  • Factor pairs: 786 = 1 x 786, 2 x 393, 3 x 262, or 6 x 131
  • 786 has no square factors that allow its square root to be simplified. √786 ≈ 28.03569.

786-factor-pairs

Today’s Find the Factors puzzle:

786-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s a little more about the number 786:

786 is 123 in BASE 27 because 1(27²) + 2(27) + 3(1) = 786.

786 is the sum of two consecutive primes: 389 + 397 = 786

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786-logic

Can You See How 779’s Factor Pairs Are Hiding in Some Pythagorean Triples?

  • 779 is a composite number.
  • Prime factorization: 779 = 19 x 41
  • 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 779 has exactly 4 factors.
  • Factors of 779: 1, 19, 41, 779
  • Factor pairs: 779 = 1 x 779 or 19 x 41
  • 779 has no square factors that allow its square root to be simplified. √779 ≈ 27.91057.

779-factor-pairs

Those factor pairs are hiding in some Pythagorean triples. Scroll down to read how, but first here’s today’s puzzle:

779 Puzzle

Print the puzzles or type the solution on this excel file: 10-factors-2016

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And now some Pythagorean triple number theory using 779 as an example:

The factors of 779 are very well hidden in five Pythagorean triples that contain the number 779. Here’s how: 779 has two factor pairs:  19 x 41 and 1 x 779. Those factor pairs show up in some way in each of the calculations for these 779 containing Pythagorean triples:

  1. 171-760-779 which is 19 times each number in 9-40-41.
  2. 779-303420-303421, a primitive calculated from 779(1); (779² – 1²)/2; (779² + 1²)/2.
  3. 779-7380-7421 which is 41 times each number in 19-180-181.
  4. 779-15960-15979 which is 19 times each number in 41-840-841.
  5. 660-779-1021, a primitive calculated from (41² – 19²)/2; 19(41); (41² + 19²)/2.

Being able to find whole numbers that satisfy the equation a² + b² = c² is one reason why finding factors of a number is so worth it. ANY factor pair for numbers greater than 2 will produce at least one Pythagorean triple that satisfies a² + b² = c². The more factor pairs a number has, the more Pythagorean triples will exist that contain that number. 779 has only two factor pairs so there are a modest number of 779 containing Pythagorean triples. All of its factors are odd so it was quite easy to find all of the triples. Here’s a brief explanation on how each triple was found:

  1. 799 has one prime factor that has a remainder of 1 when divided by 4. That prime factor, 41, is therefore the hypotenuse of a primitive Pythagorean triple. When the Pythagorean triple is multiplied by the other half of 41’s factor pair, 19, we get a Pythagorean triple in which 779 is the hypotenuse.
  2. Every odd number greater than 1 is the short leg of a primitive Pythagorean triple. To find that primitive for a different odd number, simply substitute the desired odd number in the calculation in place of 779.
  3. Because every odd number greater than 1 is the short leg of a primitive Pythagorean triple, 19(1); (19² – 1²)/2; (19² + 1²)/2 generates the primitive triple (19-180-181). Multiplying each number in that triple by the other half of 19‘s factor pair, 41, produces a triple with 779 as the short leg.
  4. Because every odd number greater than 1 is the short leg of a primitive Pythagorean triple, 41(1); (41² – 1²)/2; (41² + 1²)/2 generates the primitive triple (41-840-841). Multiplying each number in that triple by the other half of 41‘s factor pair, 19, produces a triple with 779 as the short leg.
  5. Since factor pair 19 and 41 have no common prime factors, the formula (41² – 19²)/2; 19(41); (41² + 19²)/2 produces another primitive triple 660-779-1021. If they did have common factors, the factor pair would still produce a triple, but it would not be a primitive one.

Here’s some other interesting facts about the number 779:

779 is the sum of eleven consecutive prime numbers:

47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 = 779.

779 can also be written as the sum of three squares six different ways:

  • 27² + 7² + 1² = 779
  • 27² + 5² + 5² = 779
  • 23² + 15² + 5² = 779
  • 23² + 13² + 9² = 779
  • 21² + 17² + 7² = 779
  • 21² + 13² + 13² = 779

Finally, the table below shows some logical steps that could be used to solve Puzzle #779:

779-logic

 

 

 

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