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@GWOMaths One might say that physicists study the symmetry of nature, while mathematicians study the nature of symmetry.

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@GWOMaths Observing symmetry in nature, such as noting the similarity between the symmetries of a snowflake and a hexagon, is readily comprehensible. What does it mean then to study "the nature of symmetry"?

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@GWOMaths Mathematicians define a "group", G, as a set of elements {a,b,c, ...} with a binary operation ⊡ and a distinguished element e (the identity of G) satisfying these specific properties:

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@pnjaban None of the above:

Field-trained former military paramedic.

I don't want anyone not trained specifically in emergency medicine anywhere near; and none of those civilian trained E.R. either, cause they're trained in CYA first, and I want someone who's serious about saving my life

Field-trained former military paramedic.

I don't want anyone not trained specifically in emergency medicine anywhere near; and none of those civilian trained E.R. either, cause they're trained in CYA first, and I want someone who's serious about saving my life

@pnjaban When my Dad suffered acute hemechromatosis at ~34,

for which the ONLY treatment is *bleeding*,

and the only options "a whole lot" and "a whole lot more",

the attending physician was too embarrassed to prescribe the correct treatment.

for which the ONLY treatment is *bleeding*,

and the only options "a whole lot" and "a whole lot more",

the attending physician was too embarrassed to prescribe the correct treatment.

@pnjaban The nurse (Irish of course, because they understand hemechromatosis) told me Mom:

Mrs. Geerkens:

Your husband will die tonight

unless we remove him from this hospital immediately.

I'll help.

The nurse lost her job;

but my Dad lived another 45 years.

Mrs. Geerkens:

Your husband will die tonight

unless we remove him from this hospital immediately.

I'll help.

The nurse lost her job;

but my Dad lived another 45 years.

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.

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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.

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

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@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.

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@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.