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I-have-25-horses-and-I-d-like-to-know-which-are-the-three-fastest-horses-among-them-I-do-not-have-a-clock-but-I-have-a-race-track-which-can-be-used-by-5-horses-at-a-time-If-each-horse-covers-the-dis




Question Number 3274 by Yozzi last updated on 09/Dec/15
I have 25 horses and I′d like to  know which are the three fastest  horses among them. I do not have a  clock but I have a race track which  can be used by 5 horses at a time.  If each horse covers the distance  of the track in the same time for  every race it runs, find the least  number of races that ought to be  held in order to identify the three  fastest horses.
$${I}\:{have}\:\mathrm{25}\:{horses}\:{and}\:{I}'{d}\:{like}\:{to} \\ $$$${know}\:{which}\:{are}\:{the}\:{three}\:{fastest} \\ $$$${horses}\:{among}\:{them}.\:{I}\:{do}\:{not}\:{have}\:{a} \\ $$$${clock}\:{but}\:{I}\:{have}\:{a}\:{race}\:{track}\:{which} \\ $$$${can}\:{be}\:{used}\:{by}\:\mathrm{5}\:{horses}\:{at}\:{a}\:{time}. \\ $$$${If}\:{each}\:{horse}\:{covers}\:{the}\:{distance} \\ $$$${of}\:{the}\:{track}\:{in}\:{the}\:{same}\:{time}\:{for} \\ $$$${every}\:{race}\:{it}\:{runs},\:{find}\:{the}\:{least} \\ $$$${number}\:{of}\:{races}\:{that}\:{ought}\:{to}\:{be} \\ $$$${held}\:{in}\:{order}\:{to}\:{identify}\:{the}\:{three} \\ $$$${fastest}\:{horses}.\: \\ $$$$ \\ $$
Answered by prakash jain last updated on 09/Dec/15
First 5 races with numbered results  1:  h_1  h_2  h_3  h_4  h_5   2:  h_6  h_7  h_8  h_9  h_(10)   3: h_(11)  h_(12)  h_(13)  h_(14)  h_(15)   4: h_(16)  h_(17)  h_(18)  h_(19)  h_(20)   5: h_(21)  h_(22)  h_(22)  h_(24)  h_(25)   2 elimnated from each group since they cannot  take position 1 2 3 overall.  6th Race with toppers (h_1 , h_6 , h_(11) , h_(16) , h_(21) )  Let us assume results are  h_1  h_6  h_(16)  h_(21)  h_(11)    h_(11) /h_(21)  is 4/5 among toppers all participants        of race 3 and 5 are eliminated as they are        slower than  at least 4^(th)  position  Leave selection process limited to  h_1  h_6  h_(16)  h_2  h_3  h_7  h_8  h_(17)  h_(18)   h_(16)  is at best 3^(rd)  so h_(17)  and h_(18)  eliminated.  h_1  h_6  h_(16)  h_2  h_3  h_7  h_8     h_6  is at best second so h_8  can be at best 4^(th) .  h_1  h_6  h_(16)  h_2  h_3  h_7    Leaves 6.  h_1  is first so it is clearly the first. need to  find best 2 from remainin 5  h_6  h_(16)  h_2  h_3  h_7     Race 7 to determine the 2^(nd)  and 3^(rd)   7 Races are needed.
$$\mathrm{First}\:\mathrm{5}\:\mathrm{races}\:\mathrm{with}\:\mathrm{numbered}\:\mathrm{results} \\ $$$$\mathrm{1}:\:\:\mathrm{h}_{\mathrm{1}} \:\mathrm{h}_{\mathrm{2}} \:\mathrm{h}_{\mathrm{3}} \:\mathrm{h}_{\mathrm{4}} \:\mathrm{h}_{\mathrm{5}} \\ $$$$\mathrm{2}:\:\:\mathrm{h}_{\mathrm{6}} \:\mathrm{h}_{\mathrm{7}} \:\mathrm{h}_{\mathrm{8}} \:\mathrm{h}_{\mathrm{9}} \:\mathrm{h}_{\mathrm{10}} \\ $$$$\mathrm{3}:\:\mathrm{h}_{\mathrm{11}} \:\mathrm{h}_{\mathrm{12}} \:\mathrm{h}_{\mathrm{13}} \:\mathrm{h}_{\mathrm{14}} \:\mathrm{h}_{\mathrm{15}} \\ $$$$\mathrm{4}:\:\mathrm{h}_{\mathrm{16}} \:\mathrm{h}_{\mathrm{17}} \:\mathrm{h}_{\mathrm{18}} \:\mathrm{h}_{\mathrm{19}} \:\mathrm{h}_{\mathrm{20}} \\ $$$$\mathrm{5}:\:\mathrm{h}_{\mathrm{21}} \:\mathrm{h}_{\mathrm{22}} \:\mathrm{h}_{\mathrm{22}} \:\mathrm{h}_{\mathrm{24}} \:\mathrm{h}_{\mathrm{25}} \\ $$$$\mathrm{2}\:\mathrm{elimnated}\:\mathrm{from}\:\mathrm{each}\:\mathrm{group}\:\mathrm{since}\:\mathrm{they}\:\mathrm{cannot} \\ $$$$\mathrm{take}\:\mathrm{position}\:\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{overall}. \\ $$$$\mathrm{6th}\:\mathrm{Race}\:\mathrm{with}\:\mathrm{toppers}\:\left(\mathrm{h}_{\mathrm{1}} ,\:\mathrm{h}_{\mathrm{6}} ,\:\mathrm{h}_{\mathrm{11}} ,\:\mathrm{h}_{\mathrm{16}} ,\:\mathrm{h}_{\mathrm{21}} \right) \\ $$$$\mathrm{Let}\:\mathrm{us}\:\mathrm{assume}\:\mathrm{results}\:\mathrm{are} \\ $$$$\mathrm{h}_{\mathrm{1}} \:\mathrm{h}_{\mathrm{6}} \:\mathrm{h}_{\mathrm{16}} \:\mathrm{h}_{\mathrm{21}} \:\mathrm{h}_{\mathrm{11}} \: \\ $$$$\mathrm{h}_{\mathrm{11}} /\mathrm{h}_{\mathrm{21}} \:\mathrm{is}\:\mathrm{4}/\mathrm{5}\:\mathrm{among}\:\mathrm{toppers}\:\mathrm{all}\:\mathrm{participants} \\ $$$$\:\:\:\:\:\:\mathrm{of}\:\mathrm{race}\:\mathrm{3}\:\mathrm{and}\:\mathrm{5}\:\mathrm{are}\:\mathrm{eliminated}\:\mathrm{as}\:\mathrm{they}\:\mathrm{are} \\ $$$$\:\:\:\:\:\:\mathrm{slower}\:\mathrm{than}\:\:\mathrm{at}\:\mathrm{least}\:\mathrm{4}^{{th}} \:\mathrm{position} \\ $$$$\mathrm{Leave}\:\mathrm{selection}\:\mathrm{process}\:\mathrm{limited}\:\mathrm{to} \\ $$$$\mathrm{h}_{\mathrm{1}} \:\mathrm{h}_{\mathrm{6}} \:\mathrm{h}_{\mathrm{16}} \:\mathrm{h}_{\mathrm{2}} \:\mathrm{h}_{\mathrm{3}} \:\mathrm{h}_{\mathrm{7}} \:\mathrm{h}_{\mathrm{8}} \:\mathrm{h}_{\mathrm{17}} \:\mathrm{h}_{\mathrm{18}} \\ $$$$\mathrm{h}_{\mathrm{16}} \:\mathrm{is}\:\mathrm{at}\:\mathrm{best}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{so}\:\mathrm{h}_{\mathrm{17}} \:\mathrm{and}\:\mathrm{h}_{\mathrm{18}} \:\mathrm{eliminated}. \\ $$$$\mathrm{h}_{\mathrm{1}} \:\mathrm{h}_{\mathrm{6}} \:\mathrm{h}_{\mathrm{16}} \:\mathrm{h}_{\mathrm{2}} \:\mathrm{h}_{\mathrm{3}} \:\mathrm{h}_{\mathrm{7}} \:\mathrm{h}_{\mathrm{8}} \:\: \\ $$$$\mathrm{h}_{\mathrm{6}} \:\mathrm{is}\:\mathrm{at}\:\mathrm{best}\:\mathrm{second}\:\mathrm{so}\:\mathrm{h}_{\mathrm{8}} \:\mathrm{can}\:\mathrm{be}\:\mathrm{at}\:\mathrm{best}\:\mathrm{4}^{\mathrm{th}} . \\ $$$$\mathrm{h}_{\mathrm{1}} \:\mathrm{h}_{\mathrm{6}} \:\mathrm{h}_{\mathrm{16}} \:\mathrm{h}_{\mathrm{2}} \:\mathrm{h}_{\mathrm{3}} \:\mathrm{h}_{\mathrm{7}} \: \\ $$$$\mathrm{Leaves}\:\mathrm{6}. \\ $$$$\mathrm{h}_{\mathrm{1}} \:\mathrm{is}\:\mathrm{first}\:\mathrm{so}\:\mathrm{it}\:\mathrm{is}\:\mathrm{clearly}\:\mathrm{the}\:\mathrm{first}.\:\mathrm{need}\:\mathrm{to} \\ $$$$\mathrm{find}\:\mathrm{best}\:\mathrm{2}\:\mathrm{from}\:\mathrm{remainin}\:\mathrm{5} \\ $$$$\mathrm{h}_{\mathrm{6}} \:\mathrm{h}_{\mathrm{16}} \:\mathrm{h}_{\mathrm{2}} \:\mathrm{h}_{\mathrm{3}} \:\mathrm{h}_{\mathrm{7}} \:\: \\ $$$$\mathrm{Race}\:\mathrm{7}\:\mathrm{to}\:\mathrm{determine}\:\mathrm{the}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{and}\:\mathrm{3}^{\mathrm{rd}} \\ $$$$\mathrm{7}\:\mathrm{Races}\:\mathrm{are}\:\mathrm{needed}. \\ $$

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