A Multiplication Based Logic Puzzle

Posts tagged ‘pi’

628, Tau, Pi, and Level 6

The circumference of a circle with a radius of one is approximately 6.28. That’s an important enough number that it has been given the symbol “τ ” which is pronounced “tau”. τ looks a little like half of the number π, but τ = 2π.

Some people think we should get rid of π and only use τ. Other people feel that π has been used for centuries, and there is no compelling reason to change now.

π is perfect for finding the area of a circle: Area = πr². Here’s the area of a circle using tau: Area = r²τ/2.

τ is very good for finding the circumference of a circle: Circumference = τr, but that looks strange compared to 2πr. In fact, it can be difficult to tell if τr is one character or two.

The Tau Manifesto shows angle measurements in degrees, π radians and τ radians. You might want to look at some videos, too. Some people think the τ radians are simpler because the radians correspond exactly to the fractional pieces the circumference of a circle or, get this, to the fractional pieces of a pie. (τ does that, not π.) Other people think that π radians are just as good because we’re used to them, and they correspond exactly to the area of any wedge in a unit circle or the area of any slice of pie. (Which would you rather eat the circumference or the area of a pie?)

Until I wrote this post and read the link shared in the comments, I hadn’t heard anybody say that π is better for some situations while τ is better for others. (Actually it appears that π is better except in formulas that use 2π.) Diameters and radii have co-existed peacefully for centuries. I don’t understand why π and τ can’t do the same. Here’s a great video that shows both sides of the argument.

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22² + 12² = 628.

628 is the hypotenuse of the Pythagorean triple 340-528-628. The greatest common factor of those three numbers is the same as the greatest common factor of 22² and 12².

7² + 11² + 13² + 17² = 628. Thank you Stetson.edu for that fun fact about the squares of those four consecutive prime numbers.

628 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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  • 628 is a composite number.
  • Prime factorization: 628 = 2 x 2 x 157, which can be written 628 = (2^2) x 157
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 628 has exactly 6 factors.
  • Factors of 628: 1, 2, 4, 157, 314, 628
  • Factor pairs: 628 = 1 x 628, 2 x 314, or 4 x 157
  • Taking the factor pair with the largest square number factor, we get √628 = (√4)(√157) = 2√157 ≈ 25.059928.

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628 Logic

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425 and Level 6

425 ends in 25 so it can be divided evenly by 25. If I had $4.25 all in quarters. How many quarters would I have? That’s the problem that I think of when I divide by 25. All of the factors of 425 are listed below the puzzle.

For some reason unknown to me, here in the United States, dates are ordered by month, date, and year. This rather illogical way of ordering allows us to say that today is 3-14-15, which are the first five digits of pi.  It could also be said that 3-14-15 at 9:26:53 gives the first ten digits of pi.

Logical or not, it is fun to declare today as Pi Day. Today’s puzzle celebrates those first five digits:

425 Puzzle

  • 425 is a composite number.
  • Prime factorization: 425 = 5 x 5 x 17, which can be written 425 = (5^2) x 17
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 425 has exactly 6 factors.
  • Factors of 425: 1, 5, 17, 25, 85, 425
  • Factor pairs: 425 = 1 x 425, 5 x 85, or 17 x 25
  • Taking the factor pair with the largest square number factor, we get √425 = (√25)(√17) = 5√17 ≈ 20.6155

425 and all of it factors (except 1) are hypotenuses of primitive Pythagorean triples, so 425 is the hypotenuse of several triples:

  • [87-416-425] and
  • [297-304-425] are primitives
  • [65-420-425] is [13-84-85] times 5
  • [119-408-425] is [7-24-25] times 17
  • [180-385-425] is [36-77-85] times 5
  • [200-375-425] is [8-15-17] times 25
  • [255-340-425] is [3-4-5] times 85

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

425 Logic

314 and The Pi Day of Our Lives

  • 314 is a composite number.
  • Prime factorization: 314 = 2 x 157
  • 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 314 has exactly 4 factors.
  • Factors of 314: 1, 2, 157, 314
  • Factor pairs: 314 = 1 x 314 or 2 x 157
  • 314 has no square factors that allow its square root to be simplified. √314 ≈ 17.720

On 3-14-15 at 9:26:53 we will experience the most significant pi day of our lives.

Thepidayofourlives.homestead.com  is selling wristbands, t-shirts, and baseball caps commemorating this special upcoming event at very reasonable prices.

Pi Basic Friend package: 3 wristbands @ $3.14 each = $9.42 USD

Each wristband costs only $3.14 (minimum purchase is 3 wristbands) while the t-shirts and the baseball caps are only $20 each. The website offers free shipping on everything they offer anywhere! You might just want to add one or more of these items to your Christmas wishlist!

 

 

30 and Pieces of Pi

30 is a composite number. 30 = 1 x 30, 2 x 15, 3 x 10, or 5 x 6. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Prime factorization: 30 = 2 x 3 x 5.

When 30 is a clue in the FIND THE FACTORS puzzle, use 3 x 10 or 5 x 6. Only one of those pairs will work for that clue in that particular puzzle.

A bookbinding blog with just the right artistic touch for pi expressed as π, a fraction (2 ways), or a decimal!

Playful Bookbinding and Paper Works

22/7 ? Isn’t 24/7 the ratio that we’re used to seeing around, expressing complete availability? 22/7 is a round, too, though its function is more the description of a round…. a round circle. If you’re at all up on your numbers, you’ll recognize that the fraction 22/7 is a good approximation for the ratio between a circle’s circumference and diameter, in other words, pi.  If you are already quickly losing interest and are ready to stop reading  you’re probably one of the multitudes who have suffered while studying the relationships of quantities (math). Now, why am I talking about pi? I don’t think I’m alone in being a bookbinder who has a comfortable connection with numbers: so much of what I do is about measurements, ratios and counting.

I am taking some time this year to take care of health, family, and explore some ideas that have been nipping at my heels for a very long time, which includes playing around…

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