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

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