### Sunday, March 9, 2014

#### Who had the greater share?

Arithmetical Riddle: In his 'Last will and testament', the father did "bequeath to my beloved only daughter 1/4 of my fortune, 1/3 of the remainder to my younger clever son, and 1/2 the remainder to my ambitious elder son, and the remainder to my always beautiful wife". Who had the greater share?

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### Thursday, March 6, 2014

#### Difference of squares

Every odd integer is the difference of two square integers, most in more than one way.

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### Monday, March 3, 2014

#### Sums of three squares = sum of three squares

If t1 = a1*d1 + a2*d2 - b1*c1 - b2*c2; s1 = a1*d1 - a2*d2 - b1*c1 + b2*c2; t2 = a1*b2 + c1*a2 + b1*d2 + d1*c2; s2 = a1*b2 - c1*a2 + b1*d2 - d1*c2; t3 = a1*a2 + c1*b2 + b1*c2 + d1*d2; s3 = a1*a2 - c1*b2 + b1*c2 - d1*d2; then t1*t1 + t2*t2 + s3*s3 = s1*s1 + s2*s2 + t3*t3 Illustration: Set t1 = 3*19 +...

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### Thursday, September 19, 2013

#### Linear transformation of Pythagorean triples into Pythagorean triples

For any integer q, If x^2 + y^2 = z^2 Then ((2q^2 -1)x -(2q)y + (2 q^2 )z)^2 +((2q )x -(1 )y + (2q )z)^2 =((2q^2 )x -(2q)y + (2q^2+1)z)^2 Proof: ((2q^2 -1)x -(2q)y + (2 q^2 )z)^2 +((2q )x -(1 )y + (2q )z)^2 -((2q^2 )x -(2q)y + (2q^2+1)z)^2 = ((2q^2-1)^2 + (2q)^2 - (2q^2)^2) x^2 +((2q)^2 + 1 -...

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### Tuesday, June 11, 2013

#### Generalization of Quadratic Equation

You have all seen the derivation of the quadratic formula solution to A x ^ 2 + B x + C = 0. A x y + B x + C y = D is the two variable generalization of the quadratic equation. Divide through by A x ^ 2 + (B/A) x + (C/A) = 0 x ^ 2 + (B/A) x = - (C/A) x y + (B/A) x + (C/A) y = (D/A) Complete the...

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### Friday, May 31, 2013

#### Newtonian square root algorithm

The Newtonian algorithm for calculating square root: We easily see that the square root of 49 is 7. But what about the square root of 50. We know that the square root of 50 must be slighly more than 7. 7 50/7 > sqrt(50) (7 + 50/7)/2 is closer to the square root of 50 than either 7 or 50/7. (7 +...

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### Thursday, May 30, 2013

#### sum of like powers

Sum Like Powers You have probably seen the formula for the sum of the first n positive integers, and maybe even the formula for the sum of the first n square integers and first n cube integers. There is a simple rule for progressing from a formula for the sum of the first n integers raised to a...

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### Monday, October 29, 2012

####

Pick any odd positive integer. It is the difference of two squares. For example, 55 = 8*8 - 3*3 Then we can make a number series by adding up and down as follows: 8 * 8 - 3 * 3 = 55 7 * 9 - 2 * 4 = 55 6 * 10 - 1 * 5 = 55 5 * 11 - 0 * 6 = 55 4 * 12 + 1 * 7 = 55 3 * 13 + 2 * 8 = 55 2 * 14 + 3 * 9 = 55...

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### Saturday, July 28, 2012

#### Seventy Two

Seventy Two 2*2*2*3*3 = 72 2 * (2*2*3*3) = 72 2 * 36 = 72 3 * (2*2*2*3) = 72 3 * 24 = 72 (2*2) * (2*3*3) = 72 4 * 18 = 72 (2*3) * (2*2*3) = 72 6 * 12 = 72 (2*2*2) * (3*3) = 72 8 * 9 = 72

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### Sunday, July 15, 2012

#### More about Fibonacci Numbers

Consider a generalized Fibonacci sequence, G0, G1, G2, G3, G4,..... Suppose we imagine that there is a number x such that G0 = 1 G1 = x G2 = x^2 G3 = x^3 G4 = x^4 etc Is such an x possible? The only requirement is that G2 = x^2 G2 = G0 + G1 x^2 = 1 + x Multiply through by x. x^3 = x + x^2 G3 = G1 +...

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