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@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

@GWOMaths The game Nim is played by two players alternately removing items from several piles of items. Absent physical materials, with pen and paper a game can be setup as rows of vertical lines, players stroking them out to "remove" them.

@GWOMaths Each turn a player removes 1 or more items from precisely one pile, or row.

The winner is the player who does not (or, as a variant, does) remove the last item.

Consider the setup of two piles of 2 items:

xx

xx

The player to move can be forced to lose by his opponent.

The winner is the player who does not (or, as a variant, does) remove the last item.

Consider the setup of two piles of 2 items:

xx

xx

The player to move can be forced to lose by his opponent.

@GWOMaths Note that this is true in both variants.

a) Last item loses:

-Remove one item and your opponent takes both items from the other pile;

- Take both items from one pile and your opponent removes just one from the other.

b) Last item wins:

Your opponent wins by mirroring your move.

a) Last item loses:

-Remove one item and your opponent takes both items from the other pile;

- Take both items from one pile and your opponent removes just one from the other.

b) Last item wins:

Your opponent wins by mirroring your move.