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

$${I}\:{have}\:\mathrm{25}\:{horses}\:{and}\:{I}'{d}\:{like}\:{to} \\ $$$${know}\:{which}\:{are}\:{the}\:{three}\:{fastest} \\ $$$${horses}\:{among}\:{them}.\:{I}\:{do}\:{not}\:{have}\:{a} \\ $$$${clock}\:{but}\:{I}\:{have}\:{a}\:{race}\:{track}\:{which} \\ $$$${can}\:{be}\:{used}\:{by}\:\mathrm{5}\:{horses}\:{at}\:{a}\:{time}. \\ $$$${If}\:{each}\:{horse}\:{covers}\:{the}\:{distance} \\ $$$${of}\:{the}\:{track}\:{in}\:{the}\:{same}\:{time}\:{for} \\ $$$${every}\:{race}\:{it}\:{runs},\:{find}\:{the}\:{least} \\ $$$${number}\:{of}\:{races}\:{that}\:{ought}\:{to}\:{be} \\ $$$${held}\:{in}\:{order}\:{to}\:{identify}\:{the}\:{three} \\ $$$${fastest}\:{horses}.\: \\ $$$$ \\ $$

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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