MAY 29, 2023
MAY 23, 2023
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HEY WOW I WON!!
https://fivethirtyeight.com/features/can-you-game-the-currency-exchange/
(I wish it was for a more elegant solution than a brute-force probability tree...)
(I wish it was for a more elegant solution than a brute-force probability tree...)
APRIL 14, 2023
If I make an equilateral triangle ABC with side length 1, I can assign each vertex a separate color--in my diagram A is blue, B is green, and C is red. When I then construct another such equilateral triangle BCD, D must be the same color as A (in this case blue), and this is true even if B was red and C was green. Due to this "forcing" property of the ABDC quadrilateral, I call these constructions *power rhombi*.
If we then construct another power rhombus AEGF where AEF is an equilateral triangle (E and F are red and green, not necessarily respectively) and EFG is an equilateral triangle (forcing G to be the same color as A and D), and tilt* the power rhombi off their shared vertex A in such a way that D and G are one unit away from each other, we end up with a kind of oblong pentagon that must have at least one pair of points 1 unit apart with the same color assignations.
*The triangle ADG in this pentagon is isosceles with a base 1 and legs sqrt(3), so using the law of cosines the angles ADG and AGD are about 73.22 degrees, which means angle DAG (i.e. the angle formed between the long diagonals of the power rhombi) is about 33.56 degrees.
If we then construct another power rhombus AEGF where AEF is an equilateral triangle (E and F are red and green, not necessarily respectively) and EFG is an equilateral triangle (forcing G to be the same color as A and D), and tilt* the power rhombi off their shared vertex A in such a way that D and G are one unit away from each other, we end up with a kind of oblong pentagon that must have at least one pair of points 1 unit apart with the same color assignations.
*The triangle ADG in this pentagon is isosceles with a base 1 and legs sqrt(3), so using the law of cosines the angles ADG and AGD are about 73.22 degrees, which means angle DAG (i.e. the angle formed between the long diagonals of the power rhombi) is about 33.56 degrees.
