Just Count 2

Number Theory Level 5

Let a triangle is formed whose vertices are the vertices of a cube .Find the s u m sum of number of e q u i l a t e r a l equilateral , r i g h t right a n g l e d angled and s c a l e n e scalene t r i a n g l e s triangles formed.

First part here


The answer is 80.

Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

T h e r e a r e t h r e e t y p e s o f l i n e s . 12 e d g e s , 12 d i a g o n a l s o n s i x f a c e s , 4 g r e a t d i a g o n a l s . 12 e d g e s c o m b i n e w i t h 12 + 4 = 16 o t h e r l i n e s f o r m 192 t r i a n g l e s . 12 d i a g o n a l s c o m b i n e w i t h 4 g r e a t d i a g o n a l s f o r m 48 t r i a n g l e s . T o t a l 240 t r i a n g l e s . B u t a t r i a n g l e i s f o r m e d o u t o f 3 l i n e s . ⟹ e a c h l i n e i s c o u n t e d t h r i c e . S o n u m b e r o f t r i a n g l e s a r e 1 3 ∗ 240 = 80. There~ are~ three~ types~ of~ lines.\\ 12~~edges,~~~~~~~~12~~diagonals on six faces,~~~~~~~~~4~~great~diagonals.\\ 12~~edges~combine~with~12+4=16~other~lines~form~192~triangles.\\ 12~~diagonals~combine~with~4~great~diagonals~form~48~triangles.\\ Total~240~triangles. \\ But ~a~triangle~is~formed~out~of~3~lines.~~ \implies~~each~line~is~counted~thrice.\\ So~number~ of~ triangles~are~\dfrac 1 3*240=\Large \color{#D61F06}{80}.

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