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Question Number 112533 by Aina Samuel Temidayo last updated on 08/Sep/20
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{number}\:\mathrm{of}\:\mathrm{n} \\ $$$$\mathrm{integers}\:\mathrm{to}\:\mathrm{be}\:\mathrm{selected}\:\mathrm{from} \\ $$$$\mathrm{S}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},...\mathrm{11}\right\}\:\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{difference} \\ $$$$\mathrm{of}\:\mathrm{two}\:\mathrm{of}\:\mathrm{the}\:\mathrm{n}\:\mathrm{integers}\:\mathrm{is}\:\mathrm{7}. \\ $$
Commented by Aina Samuel Temidayo last updated on 08/Sep/20
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{integers}\:\mathrm{to}\:\mathrm{be}\:\mathrm{selected}\:\mathrm{from}\:\mathrm{the}\:\mathrm{set} \\ $$$$\left\{\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{11}\right\}\:\mathrm{to}\:\mathrm{ensure}\:\mathrm{that}\:\mathrm{some}\:\mathrm{two} \\ $$$$\mathrm{of}\:\mathrm{these}\:\mathrm{numbers}\:\mathrm{sum}\:\mathrm{to}\:\mathrm{12}? \\ $$
Commented by Aina Samuel Temidayo last updated on 08/Sep/20
$$\:\mathrm{What}\:\mathrm{about}\:\mathrm{this}?@\mathrm{mr}\:\mathrm{W}\: \\ $$
Answered by mr W last updated on 08/Sep/20
$${i}\:{think}\:{now}\:{i}\:{understand}\:{what}\:{the} \\ $$$${question}\:{meant}. \\ $$$${pair}\:\mathrm{8}\:{and}\:\mathrm{1} \\ $$$${pair}\:\mathrm{9}\:{and}\:\mathrm{2} \\ $$$${pair}\:\mathrm{10}\:{and}\:\mathrm{3} \\ $$$${pair}\:\mathrm{11}\:{and}\:\mathrm{4} \\ $$$${singles}\:\mathrm{5},\mathrm{6},\mathrm{7}. \\ $$$${if}\:{we}\:{select}\:{only}\:{the}\:{singles}\:\mathrm{5},\mathrm{6},\mathrm{7}\:{and} \\ $$$${only}\:{one}\:{from}\:{each}\:{pairs},\:{e}.{g}.\:\mathrm{8},\mathrm{2},\mathrm{3},\mathrm{11}. \\ $$$${we}\:{can}\:{not}\:{ensure}\:{that}\:{two}\:{of}\:{the} \\ $$$${selected}\:{numbers}\:{have}\:{the}\:{difference} \\ $$$${of}\:\mathrm{7}.\:{but}\:{if}\:{we}\:{select}\:{one}\:{number}\:{more}, \\ $$$${then}\:{we}\:{will}\:{get}\:{at}\:{least}\:{a}\:{pair}\:{and} \\ $$$${we}\:{can}\:{ensure}\:{that}\:{there}\:{are} \\ $$$${at}\:{least}\:{two}\:{numbers}\:{which}\:{have} \\ $$$${the}\:{difference}\:\mathrm{7}. \\ $$$${that}\:{means}\:{if}\:{we}\:{select}\:{at}\:{least}\:\mathrm{8} \\ $$$${numbers},\:{there}\:{are}\:{at}\:{least}\:{two}\:{from} \\ $$$${them}\:{which}\:{have}\:{the}\:{difference}\:\mathrm{7}. \\ $$$${so}\:{the}\:{answer}\:{is}\:{n}\geqslant\mathrm{8}. \\ $$
Answered by mr W last updated on 08/Sep/20
$${such}\:{that}\:{two}\:{selected}\:{numbers}\:{have} \\ $$$${the}\:{sum}\:\mathrm{12}. \\ $$$${pair}\:\mathrm{1}\:{and}\:\mathrm{11},\:\mathrm{2}\:{and}\:\mathrm{10},\:\mathrm{3}\:{and}\:\mathrm{9}, \\ $$$$\mathrm{4}\:{and}\:\mathrm{8},\:\mathrm{5}\:{and}\:\mathrm{7} \\ $$$${singles}\:\mathrm{6} \\ $$$${if}\:{we}\:{only}\:{select}\:{the}\:{single}\:\mathrm{6}\:{and} \\ $$$${one}\:{from}\:{each}\:{of}\:{the}\:\mathrm{5}\:{pairs},\:{i}.{e}. \\ $$$${if}\:{we}\:{only}\:{select}\:\mathrm{6}\:{numbers},\:{we}\:{can} \\ $$$${not}\:{ensure}\:{that}\:{two}\:{of}\:{them}\:{have}\:{the} \\ $$$${sum}\:\mathrm{12}.\:{but}\:{if}\:{we}\:{select}\:\mathrm{7}\:{numbers}, \\ $$$${then}\:{we}\:{can}\:{ensure}.\:{so}\:{in}\:{this}\:{case} \\ $$$${n}\geqslant\mathrm{7}. \\ $$
Commented by Aina Samuel Temidayo last updated on 09/Sep/20
$$\mathrm{Thanks}. \\ $$