## A Multiplication Based Logic Puzzle

### 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 of 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.

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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#### Comments on: "628, Tau, Pi, and Level 6" (6)

1. Let them both live!

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• After reading the link in the comment below, I’ve learned that apparently physicists like τ better than mathematicians and engineers do.

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• I think this may be because 2 pi crops up a lot in in wave equations, and therefore all over quantum physics. But it’s not hard to write the 2. Even theoretical physicists aren’t usually that lazy, and if they are, they just leave out numbers from their equations altogether 🙂

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• I don’t think lazy physicists exist. I had a couple of brilliant college professors who said they started out studying physics, but it was so difficult that they switched to mathematics.

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2. There really isn’t much call for switching away from pi. The Pi Manifesto lays out well enough why not: it’d be a lot of effort for no real benefit.

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• Thank you so much for your comment. I wouldn’t have known The Pi Manifesto existed without it. Tau may be an interesting number, but I think Pi will continue and should continue to be the circle constant most people prefer.

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