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Question Number 214536 by ajfour last updated on 11/Dec/24

Commented by ajfour last updated on 11/Dec/24

$O u t o f q u e s t i o n s l a b e l l e d b y j u s t$ $p e r m u t a t i o n s o f d i g i t s o f t h e n u m b e r$ $123456 w h a t r a n k d o e s t h i s Q h o l d ?$ $I n i n c r e a s i n g o r d e r, t h a t i s .$
Commented by A5T last updated on 11/Dec/24

$\overset{______________}{x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}}$ $w h e n x_{1} = 1, t h e r e a r e 5! = 120 p e r m u t a t i o n s o f$ $(x_{2}, x_{3}, x_{4}, x_{5}, x_{6})$ $w h e n x_{1} = 2 a n d x_{2} = 1$ $f o r x_{3} = 3; t h e r e a r e 3! = 6 f o r (4, 5, 6)$ $\Rightarrow 214356 - (120 + 6 + 1) t h$ $\Rightarrow 214365 - 128 t h$ $\Rightarrow 214536 - 129 t h$
Commented by ajfour last updated on 12/Dec/24

$p r e s e n t e d v e r y n i c e l y, t h a n k y o u .$
Commented by ajfour last updated on 12/Dec/24
https://youtu.be/z2FHhUUw94A?si=WvWgazaVBulNRY9J
	Answered by mr W last updated on 12/Dec/24

	Commented by mr W last updated on 12/Dec/24

	$a = 1 + r + \sqrt{{(1 + r)}^{2} - 1^{2}} = 1 + r + \sqrt{r (r + 2)}$ $b = 1 + 1 + \sqrt{{(1 + 1)}^{2} - 1^{2}} = 2 + \sqrt{3}$ $c = r + \sqrt{r (r + 2)} + 1 + \sqrt{3}$ ${(1 + r + \sqrt{r (r + 2)})}^{2} + {(2 + \sqrt{3})}^{2} = {(1 + r + \sqrt{r (r + 2)} + \sqrt{3})}^{2}$ $2 (1 + \sqrt{3}) - \sqrt{3} (1 + r) = \sqrt{3 r (r + 2)}$ $4 \sqrt{3} (1 + \sqrt{3}) (1 + r) = 19 + 8 \sqrt{3}$ $\Rightarrow r = \frac{19 + 8 \sqrt{3}}{4 \sqrt{3} (1 + \sqrt{3})} - 1 = \frac{5 \sqrt{3} + 9}{24} \approx 0.736$
	Commented by ajfour last updated on 12/Dec/24

	$W o w!$
	Answered by A5T last updated on 11/Dec/24

	Commented by A5T last updated on 11/Dec/24

	${(B D - 1)}^{2} = 1 \times 3 \Rightarrow B D = 1 + \sqrt{3}$ $A D = A E = 1 \Rightarrow A B = 1 + 1 + \sqrt{3} = 2 + \sqrt{3}$ ${(E C - r)}^{2} = r (r + 2) \Rightarrow E C = r + \sqrt{r (r + 2)}$ $\Rightarrow A C = 1 + r + \sqrt{r (r + 2)}$ $\frac{A B \times A C}{2} = \frac{1 (A B + A C + \sqrt{{A B}^{2} + {A C}^{2}})}{2}$ $\Rightarrow (2 + \sqrt{3}) (1 + r + \sqrt{r (r + 2)})$ $= 2 + \sqrt{3} + 1 + r + \sqrt{r (r + 2)} + \sqrt{7 + 4 \sqrt{3} + 1 + r^{2} + r (r + 2) + 2 r + 2 \sqrt{r (r + 2)} + 2 r \sqrt{r (r + 2)}}$ $\Rightarrow r = \frac{9 + 5 \sqrt{3}}{24}$
	Commented by ajfour last updated on 11/Dec/24

	$A w e s o m e!$
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