Find-the-minimum-number-of-n-integers-to-be-selected-from-S-1-2-3-11-so-that-the-difference-of-two-of-the-n-integers-is-7-

Question Number 112533 by Aina Samuel Temidayo last updated on 08/Sep/20

Find the minimum number of n

integers to be selected from

S = {1, 2, 3, \dots 11} so that the difference

of two of the n integers is 7 .

Commented by Aina Samuel Temidayo last updated on 08/Sep/20

What is the minimum number of

integers to be selected from the set

{1, 2, 3, \dots, 11} to ensure that some two

of these numbers sum to 12 ?

Commented by Aina Samuel Temidayo last updated on 08/Sep/20

What about this ? @ mr W

Answered by mr W last updated on 08/Sep/20

i t h i n k n o w i u n d e r s t a n d w h a t t h e

q u e s t i o n m e a n t .

p a i r 8 a n d 1

p a i r 9 a n d 2

p a i r 10 a n d 3

p a i r 11 a n d 4

s i n g l e s 5, 6, 7 .

i f w e s e l e c t o n l y t h e s i n g l e s 5, 6, 7 a n d

o n l y o n e f r o m e a c h p a i r s, e . g . 8, 2, 3, 11 .

w e c a n n o t e n s u r e t h a t t w o o f t h e

s e l e c t e d n u m b e r s h a v e t h e d i f f e r e n c e

o f 7 . b u t i f w e s e l e c t o n e n u m b e r m o r e,

t h e n w e w i l l g e t a t l e a s t a p a i r a n d

w e c a n e n s u r e t h a t t h e r e a r e

a t l e a s t t w o n u m b e r s w h i c h h a v e

t h e d i f f e r e n c e 7 .

t h a t m e a n s i f w e s e l e c t a t l e a s t 8

n u m b e r s, t h e r e a r e a t l e a s t t w o f r o m

t h e m w h i c h h a v e t h e d i f f e r e n c e 7 .

s o t h e a n s w e r i s n ⩾ 8 .

Answered by mr W last updated on 08/Sep/20

s u c h t h a t t w o s e l e c t e d n u m b e r s h a v e

t h e s u m 12 .

p a i r 1 a n d 11, 2 a n d 10, 3 a n d 9,

4 a n d 8, 5 a n d 7

s i n g l e s 6

i f w e o n l y s e l e c t t h e s i n g l e 6 a n d

o n e f r o m e a c h o f t h e 5 p a i r s, i . e .

i f w e o n l y s e l e c t 6 n u m b e r s, w e c a n

n o t e n s u r e t h a t t w o o f t h e m h a v e t h e

s u m 12 . b u t i f w e s e l e c t 7 n u m b e r s,

t h e n w e c a n e n s u r e . s o i n t h i s c a s e

n ⩾ 7 .

Commented by Aina Samuel Temidayo last updated on 09/Sep/20

Thanks .