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

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

\dots . 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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