In a chess board number of unit squares with 1)one vertex common? 2)2 vertices common?? 3)2 sides common??


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Question Number 178454 by SLVR last updated on 16/Oct/22

$I n a c h e s s b o a r d n u m b e r o f u n i t s q u a r e s$ $w i t h 1) o n e v e r t e x c o m m o n ?$ $2) 2 v e r t i c e s c o m m o n ? ?$ $3) 2 s i d e s c o m m o n ? ?$
	Answered by SLVR last updated on 16/Oct/22

	$k i n d l y h e l p m e$
	Commented by mr W last updated on 16/Oct/22

	$n e v e r t h e l e s s w e c a n n o t a n s w e r a$ $q u e s t i o n l i k e ‘ ‘ h o w m a n y s q u a r e s$ $o n a c h e s s b o a r d h a v e a c o m m o n$ $v e r t e x ?^{″} . j u s t t h i n k a b o u t i t . i t i s n o t$ $c l e a r w h a t t h e q u e s t i o n a s k s .$
	Commented by mr W last updated on 16/Oct/22

	$i t h i n k y o u r q u e s t i o n i s n o t c l e a r .$ $a t m o s t 4 u n i t s q u a r e s h a v e a c o m m o n$ $v e r t e x . a t m o s t 2 u n i t s q u r e s h a v e 2$ $c o m m o n v e r t e x e x . a t m o s t 2 u n i t$ $s q u a r e s h a v e a c o m m o n s i d e .$
	Commented by SLVR last updated on 16/Oct/22

	$S i r . . i t i s m y l u c k . . y o u a r e a n s w e r i n g$ $. . . . i t i s n o t u n i t s q u a r e s . .$ $b u t s q a r e s o f a n y s i z e . . t y p o . .$ $k i d l y f o r g i v e ? m y m i s t a k e . .$
	Answered by Acem last updated on 16/Oct/22

	$1) O n e v e r t e x c o m m o n : o n f o u r b o a r d e d g e s$ $E a c h v e r t e x i s f o r m e d o f 2 s q u a r e s$ ${N u m}_{1 v e r . c o m} = (n - 1) \times 4_{e d g e} = 28 v e r t i c e s; n = 8$ $i f y o u w a n t n u m b e r o f s q u a r e s m u l t i p l y b y 2$ $2) V e r t i c e s c o m m o n : i n s i d e b o a r d$ $E a c h v e r t e x i s f o r m e d o f 4 s q u a r e s$ ${N u m}_{v e r t i c e s c o m} = {(n - 1)}^{2} = 49 {v e r t i c e s}_{o f e a c h 4 s q u .}$ $3) S i d e s c o m m o n : i n s i d e b o a r d$ $E a c h s i d e i s f o r m e d o f 2 s q u a r e s$ ${N u m}_{s i d e s c o m} = 2 n (n - 1) = 112 {s i d e s}_{o f e a c h 2 s q u .}$
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