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From Albert H. Beiler's

Recreations in the Theory of Numbers:

Calculate in your head:

47² = ?

96² = ?

113² = ?

179² = ?

goodreads.com/book/show/5855…

Recreations in the Theory of Numbers:

Calculate in your head:

47² = ?

96² = ?

113² = ?

179² = ?

goodreads.com/book/show/5855…

47² = (47 + 3) ⋅ (47 - 3) + 3² = 50 ⋅ 44 + 9 = 2209

96² = (96+4) ⋅ (96-4) + 4² = 100 ⋅ 92 + 16 = 9216

113² = (113+3) ⋅ (113-3) + 3² = 116⋅110 + 9

= 12760 + 9 = 12,769

179² = (179+21) ⋅ (179-21) + 21²

= 200 ⋅ 158 + (20² + 20 + 21)

= 31600 + 441 = 32041

96² = (96+4) ⋅ (96-4) + 4² = 100 ⋅ 92 + 16 = 9216

113² = (113+3) ⋅ (113-3) + 3² = 116⋅110 + 9

= 12760 + 9 = 12,769

179² = (179+21) ⋅ (179-21) + 21²

= 200 ⋅ 158 + (20² + 20 + 21)

= 31600 + 441 = 32041

The trick is from noting that from difference of squares

a² - b² = (a + b) ⋅ (a - b)

one can obtain by rearrangement

a² = (a + b) ⋅ (a - b) + b².

a² - b² = (a + b) ⋅ (a - b)

one can obtain by rearrangement

a² = (a + b) ⋅ (a - b) + b².

@GWOMaths Hint follows

@GWOMaths Rochambeau is just a fancy name for Rock-Paper-Scissors.

@GWOMaths If you're having difficulty visualizing this,

consider the situation after Game 1 as the baseline.

consider the situation after Game 1 as the baseline.

@LarrySchweikart My take:

SCOTUS must not be seen to play favourites - yet has an obligation to ensure that corruption is voided.

So wherever widespread and continuous failure to observe due process (as per State law) is seen, void all ballots from that county / counting centre.

1/

SCOTUS must not be seen to play favourites - yet has an obligation to ensure that corruption is voided.

So wherever widespread and continuous failure to observe due process (as per State law) is seen, void all ballots from that county / counting centre.

1/

@LarrySchweikart If the number of voided counties exceeds more than 1 or 2, then void the entire election for the State.

In the case of House and Senate representatives, require special elections.

2/

In the case of House and Senate representatives, require special elections.

2/

@LarrySchweikart For Presidential Electors, legislation now in effect gives the State legislature responsibility and authority to select the State's slate of Electors - as per the original Constitutional design.

3/

3/

@GWOMaths Define

y ≡ x - 16

to allow factoring the term 2¹⁶ completely.

Then 2¹⁶ + 2¹⁹ + 2ˣ

= 2¹⁶ + 2¹⁹ + 2ʸ⁺¹⁶

= 2¹⁶ . (1 + 2³ + 2ʸ)

= 2¹⁶ . (9 + 2ʸ)

Now a solution, viz y=4, is readily obvious as yielding

= 2¹⁶ . (9+16)

= 2¹⁶ . 25

= (2⁸ . 5)²

∴ x = y + 16 = 20.

y ≡ x - 16

to allow factoring the term 2¹⁶ completely.

Then 2¹⁶ + 2¹⁹ + 2ˣ

= 2¹⁶ + 2¹⁹ + 2ʸ⁺¹⁶

= 2¹⁶ . (1 + 2³ + 2ʸ)

= 2¹⁶ . (9 + 2ʸ)

Now a solution, viz y=4, is readily obvious as yielding

= 2¹⁶ . (9+16)

= 2¹⁶ . 25

= (2⁸ . 5)²

∴ x = y + 16 = 20.

@GWOMaths However - is our solution unique?

Suppose

∃ a ∈ Z

such that

9 + 2ʸ ≡ a²

Then

2ʸ = a² - 9 = (a-3) . (a+3)

and both (a-3) and (a+3) must be powers of 2.

This only occurs for

a = 5 => 2ʸ = 16 => y = 4 => x = 20.

Our solution is unique.

Suppose

∃ a ∈ Z

such that

9 + 2ʸ ≡ a²

Then

2ʸ = a² - 9 = (a-3) . (a+3)

and both (a-3) and (a+3) must be powers of 2.

This only occurs for

a = 5 => 2ʸ = 16 => y = 4 => x = 20.

Our solution is unique.

@GWOMaths Finally - as a fun calculating observation,for those who have memorized powers of two at least to 2¹⁶:

(2⁸ . 5)²

= (2⁷ . 2 . 5)²

= (2⁷ . 10)²

= (128 . 10)²

= 1280²

= 2¹⁶ . 25

= 2¹⁴ . 4 . 25

= 2¹⁴ . 100

= 16,384 . 100

= 1,638,400.

(2⁸ . 5)²

= (2⁷ . 2 . 5)²

= (2⁷ . 10)²

= (128 . 10)²

= 1280²

= 2¹⁶ . 25

= 2¹⁴ . 4 . 25

= 2¹⁴ . 100

= 16,384 . 100

= 1,638,400.

@nklym143 @GWOMaths Then I'd note:

- Generating Function f(x) for a single standard die is

x.(1+x+x²+x³+x⁴+x⁵)

= x+x²+x³+x⁴+x⁵+x⁶

= Σ xⁿ . C(n,1)

1/

- Generating Function f(x) for a single standard die is

x.(1+x+x²+x³+x⁴+x⁵)

= x+x²+x³+x⁴+x⁵+x⁶

= Σ xⁿ . C(n,1)

1/

⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ - supers

Α α - Alpha

Β β - Beta

Γ γ - Gamma

Δ δ - Delta

Ε ε - Epsilon

Ζ ζ - Zeta

Θ θ - Theta

Λ λ -Lambda

Μ μ - Mu

Ν ν - Nu

Π π - Pi

Ρ ρ - Rho

Σ σ - Sigma

Τ τ - Tau

Υ υ - Upsilon

Φ φ - Phi

Χ χ - Chi

Ψ ψ - Psi

Ω ω - Omega

≅ - congruence

∴ - therefore

Α α - Alpha

Β β - Beta

Γ γ - Gamma

Δ δ - Delta

Ε ε - Epsilon

Ζ ζ - Zeta

Θ θ - Theta

Λ λ -Lambda

Μ μ - Mu

Ν ν - Nu

Π π - Pi

Ρ ρ - Rho

Σ σ - Sigma

Τ τ - Tau

Υ υ - Upsilon

Φ φ - Phi

Χ χ - Chi

Ψ ψ - Psi

Ω ω - Omega

≅ - congruence

∴ - therefore

∈ : Element

Δ : Triangle

± : Plus/minus

× ÷ : Times & Division

≤ ≠ ≥ : Inequality

∠ : Angle

° : Degree

⊥ ∥ : Perpendicular & Parallel

~ : Similarity

≡ : Equivalence

∝ : Proportional to

∞ : Infinity

≪ ≫ : Mush less/greater than

∘ : Function composition

† * : Matrix

Δ : Triangle

± : Plus/minus

× ÷ : Times & Division

≤ ≠ ≥ : Inequality

∠ : Angle

° : Degree

⊥ ∥ : Perpendicular & Parallel

~ : Similarity

≡ : Equivalence

∝ : Proportional to

∞ : Infinity

≪ ≫ : Mush less/greater than

∘ : Function composition

† * : Matrix

More superscripts:

⁺ ⁻ ⁽ ⁾

ᵃ ᵇ ᶜ ᵈ ᵉ ᶠ ᵍ ʰ ⁱ ʲ ᵏ ˡ ᵐ ⁿ ᵒ ᵖ ʳ ˢ ᵗ ᵘ ᵛ ʷ ˣ ʸ ᶻ

ᴬ ᴮ ᴰ ᴱ ᴳ ᴴ ᴵ ᴶ ᴷ ᴸ ᴹ ᴺ ᴼ ᴾ ᴿ ᵀ ᵁ ⱽ ᵂ

ᵝ ᵞ ᵟ ᶿ ᵠ ᵡ

⁺ ⁻ ⁽ ⁾

ᵃ ᵇ ᶜ ᵈ ᵉ ᶠ ᵍ ʰ ⁱ ʲ ᵏ ˡ ᵐ ⁿ ᵒ ᵖ ʳ ˢ ᵗ ᵘ ᵛ ʷ ˣ ʸ ᶻ

ᴬ ᴮ ᴰ ᴱ ᴳ ᴴ ᴵ ᴶ ᴷ ᴸ ᴹ ᴺ ᴼ ᴾ ᴿ ᵀ ᵁ ⱽ ᵂ

ᵝ ᵞ ᵟ ᶿ ᵠ ᵡ