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Question Number 33217 by Tinkutara last updated on 13/Apr/18

Commented by Rasheed.Sindhi last updated on 13/Apr/18

N=2? (I supposed that both cordinates of mid- points be not integer.)

$$\mathrm{N}=\mathrm{2}? \\ $$$$\left(\mathrm{I}\:\mathrm{supposed}\:\mathrm{that}\:\mathrm{both}\:\mathrm{cordinates}\:\mathrm{of}\:\mathrm{mid}-\right. \\ $$$$\left.\mathrm{points}\:\mathrm{be}\:\mathrm{not}\:\mathrm{integer}.\right) \\ $$

Commented by Tinkutara last updated on 13/Apr/18

I don't have answer in my book. I found 3 points.

Commented by Rasheed.Sindhi last updated on 13/Apr/18

Any three points for example?

$$\mathrm{Any}\:\mathrm{three}\:\mathrm{points}\:\mathrm{for}\:\mathrm{example}? \\ $$

Commented by Tinkutara last updated on 14/Apr/18

(0,0),(1,3)(3,4)

$$\left(\mathrm{0},\mathrm{0}\right),\left(\mathrm{1},\mathrm{3}\right)\left(\mathrm{3},\mathrm{4}\right) \\ $$

Commented by Rasheed.Sindhi last updated on 14/Apr/18

(0,0),(1,3),(3,4)&(6,5) e=even & o=odd (e,e), (e,o) , (o,o) , (o,e) are required points. (e,e), (e,o)⇒midpoint=(((e+e)/2),((e+o)/2)) (e,e), (o,o)⇒midpoint=(((e+o)/2),((e+o)/2)) (e,e), (o,e)⇒midpoint=(((e+o)/2),((e+e)/2)) (e,o), (o,o)⇒midpoint=(((e+o)/2),((o+o)/2)) (e,o), (o,e)⇒midpoint=(((e+o)/2),((o+e)/2)) (o,o), (o,e)⇒midpoint=(((o+o)/2),((o+e)/2)) written in red are not integers. −−−−−−−−−−−−− e+e=e⇒((e+e)/2) is integer o+o=e⇒((o+o)/2) is integer e+o=o⇒((e+o)/2) is not integer o+e=o⇒((o+e)/2) is not integer

$$\left(\mathrm{0},\mathrm{0}\right),\left(\mathrm{1},\mathrm{3}\right),\left(\mathrm{3},\mathrm{4}\right)\&\left(\mathrm{6},\mathrm{5}\right) \\ $$$$\mathrm{e}=\mathrm{even}\:\&\:\mathrm{o}=\mathrm{odd} \\ $$$$\left(\mathrm{e},\mathrm{e}\right),\:\left(\mathrm{e},\mathrm{o}\right)\:,\:\left(\mathrm{o},\mathrm{o}\right)\:,\:\left(\mathrm{o},\mathrm{e}\right)\:\mathrm{are}\:\mathrm{required}\:\mathrm{points}. \\ $$$$\left(\mathrm{e},\mathrm{e}\right),\:\left(\mathrm{e},\mathrm{o}\right)\Rightarrow\mathrm{midpoint}=\left(\frac{\mathrm{e}+\mathrm{e}}{\mathrm{2}},\frac{\mathrm{e}+\mathrm{o}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{e},\mathrm{e}\right),\:\left(\mathrm{o},\mathrm{o}\right)\Rightarrow\mathrm{midpoint}=\left(\frac{\mathrm{e}+\mathrm{o}}{\mathrm{2}},\frac{\mathrm{e}+\mathrm{o}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{e},\mathrm{e}\right),\:\left(\mathrm{o},\mathrm{e}\right)\Rightarrow\mathrm{midpoint}=\left(\frac{\mathrm{e}+\mathrm{o}}{\mathrm{2}},\frac{\mathrm{e}+\mathrm{e}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{e},\mathrm{o}\right),\:\left(\mathrm{o},\mathrm{o}\right)\Rightarrow\mathrm{midpoint}=\left(\frac{\mathrm{e}+\mathrm{o}}{\mathrm{2}},\frac{\mathrm{o}+\mathrm{o}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{e},\mathrm{o}\right),\:\left(\mathrm{o},\mathrm{e}\right)\Rightarrow\mathrm{midpoint}=\left(\frac{\mathrm{e}+\mathrm{o}}{\mathrm{2}},\frac{\mathrm{o}+\mathrm{e}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{o},\mathrm{o}\right),\:\left(\mathrm{o},\mathrm{e}\right)\Rightarrow\mathrm{midpoint}=\left(\frac{\mathrm{o}+\mathrm{o}}{\mathrm{2}},\frac{\mathrm{o}+\mathrm{e}}{\mathrm{2}}\right) \\ $$$$\mathrm{written}\:\mathrm{in}\:\mathrm{red}\:\mathrm{are}\:\mathrm{not}\:\mathrm{integers}. \\ $$$$−−−−−−−−−−−−− \\ $$$$\mathrm{e}+\mathrm{e}=\mathrm{e}\Rightarrow\frac{\mathrm{e}+\mathrm{e}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{integer} \\ $$$$\mathrm{o}+\mathrm{o}=\mathrm{e}\Rightarrow\frac{\mathrm{o}+\mathrm{o}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{integer} \\ $$$$\mathrm{e}+\mathrm{o}=\mathrm{o}\Rightarrow\frac{\mathrm{e}+\mathrm{o}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{not}\:\mathrm{integer} \\ $$$$\mathrm{o}+\mathrm{e}=\mathrm{o}\Rightarrow\frac{\mathrm{o}+\mathrm{e}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{not}\:\mathrm{integer} \\ $$

Commented by Rasheed.Sindhi last updated on 14/Apr/18

If we mean by “does not have integer coordinates” both coordinates non-integer, then N=2 (e,e) , (o,o) or (e,o) , (o,e)

$$\mathrm{If}\:\mathrm{we}\:\mathrm{mean}\:\mathrm{by}\:``\mathrm{does}\:\mathrm{not}\:\mathrm{have}\:\mathrm{integer}\:\mathrm{coordinates}'' \\ $$$$\mathrm{both}\:\mathrm{coordinates}\:\mathrm{non}-\mathrm{integer},\:\mathrm{then}\:\mathrm{N}=\mathrm{2} \\ $$$$\left(\mathrm{e},\mathrm{e}\right)\:,\:\left(\mathrm{o},\mathrm{o}\right)\:\:{or}\:\:\left(\mathrm{e},\mathrm{o}\right)\:,\:\left(\mathrm{o},\mathrm{e}\right) \\ $$

Commented by Tinkutara last updated on 14/Apr/18

Thank you very much Sir! I got the answer. ��������

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